For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

Exponents and Logarithms

Exponents and Logarithms

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
I wrote the codes for some of the calculators using JavaScript, a client-side scripting language. Please use the latest Internet browsers.
The Wolfram Alpha widgets (many thanks to the developers) was used for some of the calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Story

Virologist: It is a virus.
It is highly contagious.
It is spreading so fast.

Mathematician: GOD help us!
How fast is it spreading?

Statistician: On the first day (Day $1$), two people contacted it.
On the second day (Day $2$), four people contacted it.
On the third day (Day $3$), nine people contacted it.
On the fourth day (Day $4$), sixteen people were affected.
On the fifth day (Day $5$), twenty five people were affected.
On the sixth day (Day $6$), thirty six people were affected.
On the seventh day, (Day $7$), forty nine people tested positive for the virus.

Teacher: If this trend continues, how many people are likely to be infected on the ninth day?
What type of function does this scenario represent?
What is the graph of that function called?


On the ninth day, $81$ people are likely to be infected.
This represents a Quadratic function.
The graph of a quadratic function is called a parabola.

Can we represent this information in a table?

Day, $x$ Number of People, $y$
$y = x^2$
$1$ $1$
$2$ $4$
$3$ $9$
$4$ $16$
$5$ $25$
$6$ $36$

Social Worker: Wait a minute!
We have an updated report.
Here it is:

On the first day (Day $1$), two people contacted it.
On the second day (Day $2$), four people contacted it.
On the third day (Day $3$), eight people contacted it.
On the fourth day (Day $4$), sixteen people were affected.
On the fifth day (Day $5$), thirty two people were affected.
On the sixth day (Day $6$), sixty four people were affected.
On the seventh day, (Day $7$), one hundred and twenty eight people tested positive for the virus.

Teacher: If this trend continues, how many people are likely to be infected on the ninth day?
What type of function does this scenario represent?


On the ninth day, $512$ people are likely to be infected.
This represents an Exponential function.

Can we represent this updated information in a table?

Day, $x$ Number of People, $y$
$y = 2^x$
$1$ $1$
$2$ $4$
$3$ $8$
$4$ $16$
$5$ $32$
$6$ $64$

Teacher: Do you see the difference between a Quadratic Function and an Exponential Function?
Do you see the difference between $x^2$ and $2^x$?
Students: Yes Sir/Ma'am.
Teacher: Can you mention some life scenarios of "Exponential Growth - increasing at a fast rate" and "Exponential Decay - decreasing at a fast rate"?




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Objectives

Students will:

(1.) Discuss exponents.

(2.) Discuss logarithms.

(3.) Discuss the relationship between exponents and logarithms.

(4.) Discuss the relationship between the laws of exponents and the laws of logarithms.

(5.) Simplify exponential expressions.

(6.) Solve exponential equations.

(7.) Check the solution(s) of exponential equations.

(8.) Determine the logarithms of terms without a calculator.

(9.) Simplify logarithmic expressions.

(10.) Expand logarithmic expressions.

(11.) Condense logarithmic expressions.

(12.) Solve logarithmic equations.

(13.) Check the solution(s) of logarithmic equations.

(13.) Solve applied problems involving Exponential Growth.

(14.) Solve applied problems involving Exponential Decay.

(15.) Solve applied problems involving Mathematics of Finance.

(16.) Solve applied problems in Chemistry involving pH and pOH.

(17.) Solve all other applications in several disciplines.




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Vocabulary Words

Ask students to suggest possible vocabulary words for this topic.

base, power, index, exponent, logarithm, inverses, inverse functions, laws of exponents, laws of logarithms, exponential expression, exponential equation, logarithmic expression, logarithmic equation, exponential models, logarithmic models, mathematics of finance, financial models, exponential growth, exponential decay, growth constant, decay constant, half-life, doubling time,

Definitions

Exponent tells how many times to multiply the base.

The Logarithm of a term, to a base, is the power(or index or exponent) to which the base must be raised to give the term.

Reasons for Studying Exponents and Logarithms

Some common questions often asked by students:
Why am I learning this?
What am I going to do with this?
How is this applicable to my major?
Do I need this to become ... (your profession)?
Of what use is this to me?

These are all good questions!

Let us begin by asking them this question. (Answering Questions with Questions? - yes, that's right)
Typical of Nigerians ☺ ☺ ☺
Guess what though? You do not get into trouble by answering questions with questions.
Okay, back to this question.

Exponents and Logarithms lead to Exponential Functions and Logarithmic Functions
Some of the notable applications in various field/disciplines of life include:

Applications of Exponential Functions

(1.) Exponential Growth (Word Problems - Part 1)
Some examples are seen in:
Population growth - Majors in Family Studies, etc.
Continuous Compound Interest - Majors in Finance, Business, Accountancy, etc.
Epidemiology, Virology, Bacteriology, - Majors in Health Science, Medicine, Nursing, etc.

(2.) Exponential Decay (Word Problems - Part 2)
Some examples are seen in:
Radioactivity - Majors in Nuclear Physics, Nuclear Chemistry, etc.
Drug Effects - Majors in Pharmacy etc.

(3.) Mathematics of Finance (Word Problems - Part 3)
Mathematics of Finance Calculator
Some examples are seen in:
Compound Interest, Annual Percentage Yield (APY), Annuities, Amortizations, Sinking Funds, etc. - Majors in Finance, Business, Accountancy, etc.


Applications of Logarithmic Functions

(1.) Acidity and Alkalinity of solutions (Word Problems - Part 4)
Some examples are seen in:
pH and pOH (Acidity and Alkalinity of solutions) - Majors in Chemistry, Soil Science, Plant/Crop Science, Agricultural Science, Science Lab Technology, etc.

(2.) Intensity of Earthquakes (Word Problems - Part 5)
Some examples are seen in:
Ritcher scale:Measurement of earthquakes - Majors in Geology, Geosciences, Meteorology, Physics, etc.




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Exponents and Logarithms

Exponent
$2^4 = 16$
$ base = 2 \\ exponent = 4 \\ result = 16 $
As you see, every exponent has a base.
The exponent, $4$ tells you to multiply the base, $2$; four times.
The exponent tells you how many times to multiply the base.

Logarithm
$\log_2{16} = 4$
$ base = 2 \\ logarithm = 16 \\ result = 4 $
Ask students if they see any relationship.
Ask students to compare Exponents and Logarithms
Ask students to contrast Exponents and Logarithms

As you see, every logarithm also has a base.
The base, $2$ you saw in Exponent is still the base, $2$ you see in Logarithm.
The result, $16$ you see in Exponent is the logarithm, $16$ you see in Logarithm.
The exponent, $4$ you see in Exponent is the result, $4$ you see in Logarithm.

Every Logarithm has a base.
The base of the logarithm is always written as a subscript.
The base of a logarithm can be written numerically (in numbers) or in words.
For example: $\log_2{16} = \log_{two}{16}$
If the base is not specified, that means the logarithm is in base $10$
For example: $\log{100} = \log_{10}{100} = \log_{ten}{100}$

We have two main types of Logarithms
(1.) Common Logarithm. This is the logarithm to base ten.
An example is $\log_{10}{100}$

(2.) Natural or Napierian Logarithm. This is the logarithm to base $e$.
An example is $\log_e{100}$
$\log_e{100}$ can also be written as $\ln{100}$
$\log_e{x}$ can also be written as $\ln{x}$
where $e$ is known as the Euler Number or Napier's constant.

Students may ask the meaning of $e$.
Using a scientific/graphing calculator, show the students the value of $e$
They may also ask the meaning of $\ln$. Please be sure to pronounce it well for them.
For the very curious students, please direct them to Continuous Compound Interest so they understand how the value of $e$ was found.

Back to our discussion for those two cases regarding Exponents and Logarithms, the base is the same.
Let us explain this in another perspective.
Can we use the terms, input and output?
Ask students again if they see any relationship(s). Note their responses. Provide feedback.

Let us observe this relationship.

Exponents Logarithms
$2^4 = 16$ $\log_2{16} = 4$
$5^3 = 125$ $\log_5{125} = 3$
$p^c = d$ $\log_p{d} = c$
$2^{-4} = \dfrac{1}{2^4} = \dfrac{1}{16}$ $\log_2{\dfrac{1}{16}} = -4$
$243$$\dfrac{1}{5}$ = $3$ $\log_{243}{3} = \dfrac{1}{5}$
$p^c = 7$ $\log_p{7} = c$
$e^{12} = p$ $ \log_e{p} = 12 \\[3ex] OR \\[3ex] \ln {p} = 12 $
$10^{-4} = 0.0001$ $ \log_{10}{0.0001} = -4 \\[3ex] OR \\[3ex] \log {0.0001} = -4 $
$p^{-c} = d$ $\log_p{d} = -c$
$p^c = d^e$ $\log_p{d^e} = c$

Can we have a negative base for Logarithms?
That is a good question.
Let us study these examples.
$3^2 = 9$
$\log_3{9} = 2$

$(-3)^2 = 9$
***$\log_{-3}{9} = 2$***

Wait a minute!
There is a contradiction here.
We just cannot have two different bases giving us the same result of the logarithm of the same number.
Why is that? ☺ ☺ ☺ (Speaking like an American lol)

Calm down. Let us look at further examples. Then, it will make sense.
Remember: we are only dealing with real numbers. We are not interested in complex numbers.
$4$$\frac{1}{2}$ = $2$
$\log_4{2} = \frac{1}{2}$

$(-4)$$\frac{1}{2}$ = $\sqrt{-4} = 2i$
This is not a real number
***$\log_{-4}{2}$*** will not work for real numbers.

Do you see why we cannot have negative bases?

So, please note:
For Exponents
(1.) Base can be positive, zero, or negative
(2.) Exponent can be positive, zero, or negative.
(3.) Result can be positive, zero, or negative.

For Logarithms
(1.) Base can only be positive
(2.) Number can only be positive.
(3.) Result can be positive, zero, or negative.




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Laws of Exponents and Laws of Logarithms

Laws of Exponents Laws of Logarithms
(1.)
$ p^c * p^d = p^{c + d} \\[3ex] p^{c + d} = p^c * p^d \\[3ex] 3^3 * 3^4 = 3^{3 + 4} = 3^7 \\[3ex] 3^{-3} * 3^4 = 3^{-3 + 4} = 3^1 \\[3ex] 3^3 * 3^{-4} = 3^{3 + (-4)} = 3^{3 - 4} = 3^{-1} \\[3ex] 3^{-3} * 3^{-4} = 3^{-3-4} = 3^{-7} \\[3ex] (-3)^3 * (-3)^4 = (-3)^{3 + 4} = (-3)^7 \\[3ex] (-3)^{-3} * (-3)^4 \\[3ex] = (-3)^{-3 + 4} = (-3)^1 \\[3ex] (-3)^3 * (-3)^{-4} = (-3)^{3 + (-4)} \\[3ex] = (-3)^{3 - 4} = (-3)^{-1} \\[3ex] (-3)^{-3} * (-3)^{-4} \\[3ex] = (-3)^{-3-4} = (-3)^{-7} \\[3ex] $ ACT $8x^5 * 12x^5$ is equivalent to:

$ 8 * x^5 * 12 * x^5 8 * 12 * x^5 * x^5 \\[3ex] = 96 * x^{5 + 5} \\[3ex] = 96x^{10} $
(1.)
$ \log_p{c} + \log_p{d} = \log_p{cd} \\[3ex] \log_p{cd} = \log_p{c} + \log_p{d} \\[3ex] \log_7{3} + \log_7{4} \\[3ex] = \log_7{3 * 4} = \log_7{12} \\[3ex] \log{12} = \log{3} + \log{4} \\[3ex] \log_e{12} = \log_e{6} * \log_e{2} \\[3ex] \log_3{12} = \log_3{1} * \log_3{12} $
(2.)
$ p^c \div p^d = p^{c - d} \\[3ex] \dfrac{p^c}{p^d} = p^{c - d} \\[3ex] p^{c - d} = p^c \div p^d \\[3ex] p^{c - d} = \dfrac{p^c}{p^d} $
(2.)
$ \log_p{c} - \log_p{d} = \log_p({c \div d}) \\[3ex] \log_p{c} - \log_p{d} = \log_p{\dfrac{c}{d}} \\[3ex] \log_p({c \div d}) = \log_p{c} - \log_p{d} \\[3ex] \log_p{\dfrac{c}{d}} = \log_p{c} - \log_p{d} $
(3.)
$any\: base$$0$ = $1$

$ p^0 = 1 \\[3ex] (AWC)^0 = 1 \\[3ex] 5^0 = 1 \\[3ex] (-5)^0 = 1 $
(3.)
$\log_{any\: base}{1} = 0$

$ \log_p{1} = 0 \\[3ex] \log_{AWC}{1} = 0 \\[3ex] \log_3{1} = 0 \\[3ex] \ln{1} = \log_e{1} = 0 $
(4.)
$any\: base$$1$ = $any\: base$

$ p^1 = p \\[3ex] (AWC)^1 = AWC \\[3ex] 5 = 5^1 \\[3ex] (-5)^1 = -5 $
(4.)
$\log_{any\: base}{any\: base} = 1$

$ \log_p{p} = 1 \\[3ex] \log_{AWC}{AWC} = 1 \\[3ex] \log_3{3} = 1 \\[3ex] \log{10} = \log_{10}{10} = 1 $
(5.)
$ (p^c)^d = p^{c * d} \\[5ex] p^{c * d} = (p^c)^d \\[5ex] (pk)^d = p^d * k^d \\[5ex] p^d * k^d = (pk)^d \\[5ex] (p^c k^d)^m = p^{cm} * k^{dm} \\[5ex] p^{cm} * k^{dm} = (p^c)^m * (k^d)^m = (p^c k^d)^m \\[5ex] $ $(p^c)$$\dfrac{d}{e}$ = $p$$\dfrac{cd}{e}$


$p$$\dfrac{cd}{e}$ = $(p^c)$$\dfrac{d}{e}$
(5.)
$ \log_p{c^d} = d * \log_p{c} \\[3ex] d * \log_p{c} = \log_p{c^d} \\[5ex] \log_7{7^{12}} = 12 * \log_7{7} \\[3ex] = 12 * 1 = 12 \\[5ex] \log_3{9} = \log_3{3^2} \\[3ex] = 2 * \log_3{3} = 2 * 1 = 2 \\[5ex] \log_3{\sqrt{3}} = \log_3{3^{\dfrac{1}{2}}} \\[3ex] = \dfrac{1}{2} * \log_3{3} = \dfrac{1}{2} * 1 = \dfrac{1}{2} \\[5ex] \log_6{216} = \log_6{6^3} \\[3ex] = 3 * \log_6{6} = 3 * 1 = 3 \\[5ex] \ln {\sqrt{e}} = \log_e{\sqrt{e}} \\[3ex] = \log_e{e^\dfrac{1}{2}} = \dfrac{1}{2} * \log_e{e} \\[5ex] = \dfrac{1}{2} * 1 = \dfrac{1}{2} \\[5ex] \log100 = \log_{10}{100} \\[3ex] = \log_{10}{10^2} \\[3ex] = 2 * \log_{10}{10} = 2 * 1 = 2 \\[5ex] \ln e^4 = \log_e{e^4} \\[3ex] = 4 * \log_e{e} = 4 * 1 = 4 \\[5ex] \log_3{\dfrac{1}{27}} = \log_3{27^{-1}} \\[3ex] = \log_3{3^{3(-1)}} = \log_3{3^{-3}} \\[3ex] = -3 * \log_3{3} = -3 * 1 \\[3ex] = -3 \\[5ex] \log_7{\sqrt[3]{7}} = \log_7{7^{\dfrac{1}{3}}} \\[3ex] = \dfrac{1}{3} * \log_7{7} \\[3ex] = \dfrac{1}{3} * 1 = \dfrac{1}{3} \\[5ex] \ln {\dfrac{1}{e^2}} = \log_e{\dfrac{1}{e^2}} \\[3ex] = \log_e{e^{-2}} = -2 * \log_e{e} \\[3ex] = -2 * 1 = -2 \\[5ex] \log{\dfrac{1}{10}} = \log_{10}{\dfrac{1}{10}} \\[5ex] = \log_{10}{10^{-1}} = -1 * \log_{10}{10} \\[3ex] = -1 * 1 = -1 $
(6.)
$ p^{-c} = \dfrac{1}{p^c} \\[3ex] \dfrac{1}{p^c} = p^{-c} \\[3ex] p^c = \dfrac{1}{p^{-c}} \\[3ex] \dfrac{1}{p^{-c}} = p^c \\[3ex] $
$p$$-{\dfrac{c}{d}}$ = $\dfrac{1}{p^{\dfrac{c}{d}}}$

$\dfrac{1}{p^{\dfrac{c}{d}}}$ = $p$$-{\dfrac{c}{d}}$

$p$${\dfrac{c}{d}}$ = $\dfrac{1}{p^-{\dfrac{c}{d}}}$

$\dfrac{1}{p^-{\dfrac{c}{d}}}$ = $p$${\dfrac{c}{d}}$
(6.)
Change of Base of Log
$ \log_p{d} = \dfrac{\log_c{d}}{\log_c{p}} \\[3ex] \dfrac{\log_c{d}}{\log_c{p}} = \log_p{d} \\[3ex] \log_p{d} * \log_c{p} = \log_c{d} \\[3ex] \log_c{d} = \log_p{d} * \log_c{p} \\[3ex] \log_{216}{6} = \dfrac{\log_6{6}}{\log_6{216}} \\[3ex] = \dfrac{1}{3} \\[5ex] \log_{\sqrt{3}}{9} = \dfrac{\log_3{9}}{\log_3{\sqrt{3}}} \\[5ex] \log_3{9} = \log_3{3^2} = 2 \\[3ex] \log_3{\sqrt{3}} = \log_3{3^{\dfrac{1}{2}}} = \dfrac{1}{2} \\[5ex] = 2 \div \dfrac{1}{2} \\[5ex] = 2 * \dfrac{2}{1} \\[5ex] = 4 $
(7.)

$p$$\dfrac{1}{c}$ = $\sqrt[c]{p}$

$p$$\dfrac{c}{d}$ = $\sqrt[d]{p^c}$

$p$$\dfrac{c}{d}$ = $(\sqrt[d]{p})^c$

$\sqrt[d]{p^c}$ = $p$$\dfrac{c}{d}$

$(\sqrt[d]{p})^c$ = $p$$\dfrac{c}{d}$

$p$$\dfrac{1}{2}$ = $\sqrt{p}$

$\sqrt{p}$ = $p$$\dfrac{1}{2}$

$p$$\dfrac{1}{3}$ = $\sqrt[3]{p}$

$\sqrt[3]{p}$ = $p$$\dfrac{1}{3}$

$p$$\dfrac{3}{4}$ = $\sqrt[4]{p^3}$

$p$$\dfrac{3}{4}$ = $(\sqrt[4]{p})^3$

$p$$\dfrac{4}{3}$ = $\sqrt[3]{p^4}$

$p$$\dfrac{4}{3}$ = $(\sqrt[3]{p})^4$

(7.)

$p$$\log_p{c}$ = $c$

$c$ = $p$$\log_p{c}$

$p$$d\log_p{c}$ = $p$$\log_p{c^d}$ = $c^d$

$c^d$ = $p$$\log_p{c^d}$ = $p$$d\log_p{c}$

$e$$\ln{c}$ = $c$

$c$ = $e$$\ln{c}$

$e$$d\ln{c}$ = $e$$\ln{c^d}$ = $c^d$

$c^d$ = $e$$\ln{c^d}$ = $e$$d\ln{c}$

$2$$\log_2{3}$ = $3$

$e$$7\ln{x}$ = $x^7$



Ask students to prove Law 7 of Logarithms based on the relationship between exponents and logarithms.

Can we write these laws in English Language?

Laws of Exponents

Law 1:
If two expressions have: the same base, and are being multiplied;
Keep the base
Add the exponents.

Law 2:
If two expressions have: the same base, and are being divided;
Keep the base
Subtract the exponents.

Law 3:
Any base raised to an exponent of $0$ gives a result of $1$
In other words, the result of any base raised to an exponent of $0$, is $1$

Law 4:
Any base raised to an exponent of $1$ is that base.
In other words, the result of any base raised to an exponent of $1$, is that base.

Law 5:
If an expression enclosed in parenthesis has two exponents: an inner exponent and an outer exponent;
Keep the base
Multiply the exponents.

Law 6:
A base with a negative exponent is the reciprocal of the same base with the corresponding positive exponent.
A base with a positive exponent is the reciprocal of the same base with the corresponding negative exponent.

Law 7:
Fractional exponents leads to radicals.
In other words, a base whose exponent is a fraction can be expressed as a radical.

Laws of Logarithms

Law 1:
If two logarithms have: the same base, and are being added;
Keep the base
Multiply the numbers.

Law 2:
If two logarithms have: the same base, and are being subtracted;
Keep the base
Divide the numbers.

Law 3:
The logarithm of the number, $1$; to any base gives a result of $0$
In other words, the result of the logarithm of $1$; to any base is $0$

Law 4:
The logarithm of a number; to that number as the base, gives a result of $1$
In other words, the result of the logarithm of a number to that same number as the base is $1$

Law 5:
The logarithm of a number to a base, raised to an exponent; is equal to the exponent times the logarithm of that number to the base.

Law 6:
This Law deals with the Change of Base of Logarithms.
Any logarithm of a number to a base, can be expressed as a ratio of two logarithms to another base.
Say you have the logarithm of a number say $d$ to a base, say $p$; and
You want to change the base of that logarithm to another base, say $c$; then the logarithm of $d$ to base $p$ is the logarithm of $d$ to base $c$ divided by the logarithm of $p$ to base $c$.
In other words, the logarithm of $d$ to base $p$ is the ratio of the logarithm of $d$ to base $c$, to the logarithm of $p$ to base $c$

Law 7:
This Law actually deals with both exponents and logarithms.
The result of a base, raised to an exponent of the logarithm of a number whose base is the base, is the number.


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Exponential/Logarithmic Expressions Calculator

For Exponential Expressions;
This calculator will:
(1.) Simplify exponential expressions.
(2.) Give the answer in terms of positive exponents as applicable.
(3.) Display the 3D (3-dimensional) plot of the solution as applicable.
(4.) Display the Contour plot of the solution as applicable.

For Logarithmic Expressions;
This calculator will:
(1.) Simplify logarithmic expressions.
(2.) Give the answer in terms of positive exponents as applicable.
(3.) Display the 3D (3-dimensional) plot of the solution as applicable.
(4.) Display the Contour plot of the solution as applicable.

To use the calculator, please:
(1.) Type your expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" expression in the textbox of the calculator.
(4.) Copy and paste the expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct expression you typed.
For Exponential Expressions; you will see that expression in terms of positive exponent(s) if it has any negative exponent(s).
(7.) Review the answer (Exponential Expression).
(8.) Review the answers (Logarithmic Expressions). At least one of the answers is what you need.

  • Using the Exponential/Logarithmic Expressions Calculator
  • Exponential Expressions: Type: $5^3 * 5^{-3}$ as 5^3 * 5^(-3)
  • Exponential Expressions: Type: $x^3 * x^{-3}$ as x^3 * x^(-3)
  • Exponential Expressions: Type: $p^{-7} * x^{-4}$ as p^(-7) / p^(-4)
  • Exponential Expressions: Type: $(d^3 * e^{-3} * f^{-2})^5$ as (d^3 * e^(-3) * f^(-2))^5
  • Exponential Expressions: Type: $(5d^3 * e^{-3} * f^{-2})^{-5}$ as (5 * d^3 * e^(-3) * f^(-2))^(-5)
  • Exponential Expressions: Type: $\left(\dfrac{a^{2}}{c^{-3}}\right)^{-2}$ as (a^2 / c^(-3))^(-2)
  • Exponential Expressions: Type: $(-7x^2y^{-4})(-x^{-3}y^7)$ as (-7 * x^2 * y^(-4))(-1 * x^(-3) * y^7)
  • Exponential Expressions: Type: $\dfrac{-48c^{-2}d^{-3}}{4c^{-3}d^{-1}}$ as (-48 * c^(-2) * d^(-3)) / (4 * c^(-3) * d^(-1))
  • Logarithmic Expressions: Type: $\log 10$ as log_10(10)
  • Logarithmic Expressions: Type: $\log_e{e}$ as log e
  • Logarithmic Expressions: Type: $\ln e$ as log e
  • Logarithmic Expressions: Type: $\log{100} - \log{10}$ as log_10(100) - log_10(10)
  • Logarithmic Expressions: Type: $\log_e{100} - \log_e{10}$ as log(100) - log(10)

Solve




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Exponential/Logarithmic Equations Calculator

This calculator will:
(1.) Solve one-variable exponential equations.
(2.) Solve one-variable logarithmic equations.
(3.) Give the answer(s) in the simplest exact forms.
(4.) Graph the real solution(s) on a number line.
To see the answer(s) in decimals, click the "Approximate forms" link.
To see the answer(s) in the simplest / exact forms, click the "Exact forms" link.

To use the calculator, please:
(1.) Type your equation in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" expression in the textbox of the calculator.
(4.) Copy and paste the equation you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct equation you typed.
(7.) Review the answers.

  • Using the Exponential/Logarithmic Equations Calculator

Solve



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Logarithms Calculator - TI-84/84 Plus

This calculator will:
(1.) Solve logarithms of numbers only (not variables) to any base.

To use the calculator, please:


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References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Coburn, J., & Coffelt, J. (2014). College Algebra Essentials ($3^{rd}$ ed.). New York: McGraw-Hill

Kaufmann, J., & Schwitters, K. (2011). Algebra for College Students (Revised/Expanded ed.). Belmont, CA: Brooks/Cole, Cengage Learning.

Lial, M., & Hornsby, J. (2012). Beginning and Intermediate Algebra (Revised/Expanded ed.). Boston: Pearson Addison-Wesley.

Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.

Alpha Widgets Overview Tour Gallery Sign In. (n.d.). Retrieved from http://www.wolframalpha.com/widgets/

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/

VAST LEARNERS. (n.d). Retrieved from https://www.vastlearners.com/free-jamb-past-questions/


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