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# Expand and Condense Logarithms

To Expand Logarithms is to write a single logarithmic expression as several logarithmic expressions.

To Condense Logarithms is to write several logarithmic expressions as a single logarithmic expression.

We expand and condense logarithms using the laws of logarithms and the laws of exponents.

For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

For JAMB, NZQA, and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE:FM is a question for the WASSCE Further Mathematics/Elective Mathematics

For GCSE Students
All work is shown to satisfy (and actually exceed) the minimum for awarding method marks.
Calculators are allowed for some questions. Calculators are not allowed for some questions.

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Expand or Condense these logarithms.
If it is a single logarithmic expression, expand it.
If there are several logarithmic expressions, condense them.
Indicate any law that you used.
Show all work.

(1.) $\log_t{36} + \log_t{19}$

$\log_t{36} + \log_t{19} \\[3ex] = \log_t{36 * 19} ...Law\: 1...Log \\[3ex] = \log_t{684}$
(2.) $\ln (pc)$

$\ln (pc) \\[3ex] = \ln p + \ln c ...Law\: 1 ... Log$
(3.) $2\log_{p}{c} - 3\log_{p}{d}$

$2\log_{p}{c} - 3\log_{p}{d} \\[3ex] = \log_{p}{c^2} - \log_{p}{d^3} ...Law\: 5...Log \\[3ex] = \log_{p}{\left(\dfrac{c^2}{d^3}\right)} ...Law\:2...Log$
(4.) $\log_3{p^{-7}}$

$\log_3{p^{-7}} \\[3ex] = -7\log_3{p} ...Law\: 5 ... Log$
(5.) $\log_2{(8x)}$

$\log_2{(8x)} \\[3ex] = \log_2{(2^3 * x)} \\[3ex] = \log_2{2^3} + \log_2{x} ...Law\: 1...Log \\[3ex] \log_2{2^3} = 3\log_2{2} ...Law\: 5...Log \\[3ex] \log_2{2} = 1 ...Law\: 4...Log \\[3ex] = 3\log_2{2} + \log_2{x} \\[3ex] = 3 * 1 + \log_2{x} \\[3ex] = 3 + \log_2{x}$
(6.) $\ln \sqrt[3]{7}$

$\ln \sqrt[3]{7} \\[3ex] \sqrt[3]{7} = 7^{\dfrac{1}{3}} ...Law\: 7...Exp \\[5ex] = \ln 7^{\dfrac{1}{3}} \\[5ex] = \dfrac{1}{3} \ln 7 ...Law\: 5...Log$
(7.) $\log_p{c} + \log_p{d} - \log_p{e}$

$\log_p{c} + \log_p{d} - \log_p{e} \\[3ex] \log_p{c} + \log_p{d} = \log_p{cd} ...Law\: 1...Log \\[3ex] \log_p{cd} - \log_p{e} = \log_p{\left(\dfrac{cd}{e}\right)} ...Law\: 2...Log$
(8.) $\ln (ex)$

$\ln (ex) \\[2ex] = \ln e + \ln x ...Law\: 1...Log \\[2ex] \ln e = \log_e{e} = 1 ...Law\: 4...Log \\[2ex] = 1 + \ln x$
(9.) $\log_p{c} - (\log_p{d} - \log_p{e})$

$\log_p{c} - (\log_p{d} - \log_p{e}) \\[3ex] = \log_p{c} - \log_p{d} + \log_p{e} \\[3ex] \log_p{c} - \log_p{d} = \log_p{\left(\dfrac{c}{d}\right)} ...Law\: 2...Log \\[5ex] = \log_p{\left(\dfrac{c}{d}\right)} + \log_p{e} = \log_p{\left(\dfrac{ce}{d}\right)} ...Law\: 1...Log \\[5ex] OR \\[3ex] \log_p{c} - (\log_p{d} - \log_p{e}) \\[3ex] PEMDAS \\[3ex] \log_p{d} - \log_p{e} = \log_p{\dfrac{d}{e}} ...Law\: 2...Log \\[5ex] \log_p{c} - \log_p{\dfrac{d}{e}} = \log_p{c \div \dfrac{d}{e}} ...Law\: 2...Log \\[5ex] = \log_p{\left(c * \dfrac{e}{d}\right)} \\[5ex] = \log_p{\left(\dfrac{ce}{d}\right)}$
(10.) $\log_p{(c^3d^7e^{10})}$

$\log_p{(c^3d^7e^{10})} \\[3ex] = \log_p{c^3} + \log_p{d^7} + \log_p{e^{10}} ...Law\: 1...Log \\[3ex] = 3\log_p{c} + 7\log_p{d} + 10\log_p{e} ...Law\: 5...Log$
(11.) $(\log_p{c} - \log_p{d}) - \log_p{e}$

$(\log_p{c} - \log_p{d}) - \log_p{e} \\[3ex] (\log_p{c} - \log_p{d}) = \log_p{\left(\dfrac{c}{d}\right)} ...Law\: 2...Log \\[5ex] = \log_p{\left(\dfrac{c}{d}\right)} - \log_p{e} = \log_p{\left(\dfrac{c}{d} \div e \right)} ...Law\: 2...Log \\[5ex] \log_p{\left(\dfrac{c}{d} \div e \right)} = \log_p{\left(\dfrac{c}{d} * \dfrac{1}{e} \right)} \\[5ex] = \log_p{\left(\dfrac{c}{d} * \dfrac{1}{e} \right)} \\[5ex] = \log_p{\left(\dfrac{c}{de}\right)}$
(12.) $\ln {\left(\dfrac{x}{e^x}\right)}$

$\ln {\left(\dfrac{x}{e^x}\right)} \\[5ex] = \ln x - \ln {e^x} ...Law\: 2...Log \\[3ex] \ln {e^x} = x\ln e = 1 ...Law\: 5...Log \\[3ex] \ln e = \log_e{e} = 1 ...Law\: 4...Log \\[3ex] = \ln x - x\ln e \\[3ex] = \ln x - (x * 1) \\[3ex] = \ln x - x$
(13.) $\log_p{\sqrt[4]{\dfrac{c^{12}d^{16}}{p^3e^5}}}$

$\log_p{\sqrt[4]{\dfrac{c^{12}d^{16}}{p^3e^5}}} \\[7ex] \sqrt[4]{\dfrac{c^{12}d^{16}}{p^3e^5}} = \left(\dfrac{c^{12}d^{16}}{p^3e^5}\right)^{\dfrac{1}{4}} ...Law\: 7...Exp \\[7ex] \left(\dfrac{c^{12}d^{16}}{p^3e^5}\right)^{\dfrac{1}{4}} = \dfrac{c^{12 * \dfrac{1}{4}}d^{16 * \dfrac{1}{4}}}{p^{3 * \dfrac{1}{4}}e^{5 * \dfrac{1}{4}}} ...Law\: 5...Exp \\[7ex] \dfrac{c^{12 * \dfrac{1}{4}}d^{16 * \dfrac{1}{4}}}{p^{3 * \dfrac{1}{4}}e^{5 * \dfrac{1}{4}}} = \dfrac{c^3d^4}{p^{\dfrac{3}{4}}e^{\dfrac{5}{4}}} \\[7ex] = \log_p{\left(\dfrac{c^3d^4}{p^{\dfrac{3}{4}}e^{\dfrac{5}{4}}}\right)} \\[7ex] = \log_p{(c^3d^4)} - \log_p{(p^{\dfrac{3}{4}}e^{\dfrac{5}{4}})} ...Law\: 2...Log \\[3ex] \log_p{(c^3d^4)} = \log_p{c^3} + \log_p{d^4} ...Law\: 1...Log \\[3ex] \log_p{(p^{\dfrac{3}{4}}e^{\dfrac{5}{4}})} = \log_p{p^{\dfrac{3}{4}}} + \log_p{e^{\dfrac{5}{4}}} ...Law\: 1...Log \\[3ex] \log_p{c^3} = 3\log_p{c} ...Law\: 5...Log \\[3ex] \log_p{d^4} = 4\log_p{d} ...Law\: 5...Log \\[5ex] \log_p{p^{\dfrac{3}{4}}} = \dfrac{3}{4}\log_p{p} ...Law\: 5...Log \\[3ex] \log_p{p} = 1 ...Law\: 4...Log \\[3ex] \log_p{p^{\dfrac{3}{4}}} = \dfrac{3}{4}\log_p{p} = \dfrac{3}{4} * 1 = \dfrac{3}{4} \\[5ex] \log_p{e^{\dfrac{5}{4}}} = \dfrac{5}{4}\log_p{e} ...Law\: 5...Log \\[3ex] = \log_p{c^3} + \log_p{d^4} - (\log_p{p^{\dfrac{3}{4}}} + \log_p{e^{\dfrac{5}{4}}}) \\[5ex] = \log_p{c^3} + \log_p{d^4} - \log_p{p^{\dfrac{3}{4}}} - \log_p{e^{\dfrac{5}{4}}} \\[5ex] = 3\log_p{c} + 4\log_p{d} - \dfrac{3}{4} - \dfrac{5}{4}\log_p{e}$
(14.) $\ln {\left(\dfrac{2}{5x^5y}\right)}$

$\ln {\left(\dfrac{2}{5x^5y}\right)} \\[3ex] = \ln{2} - \ln{5x^5y} ...Law\: 2...Log \\[2ex] \ln{5x^5y} = \ln{5} + \ln{x^5} + \ln{y} ...Law\: 1...Log \\[2ex] \ln{x^5} = 5\ln{x} ...Law\: 5...Log \\[2ex] = \ln{2} - (\ln{5} + 5\ln{x} + \ln{y}) \\[2ex] = \ln{2} - \ln{5} - 5\ln{x} - \ln{y}$
(15.) $\ln{\left(\dfrac{x}{x + 3}\right)} + \ln{\left(\dfrac{x - 3}{x}\right)} - \ln{(x^2 - 9)}$

$\ln{\left(\dfrac{x}{x + 3}\right)} + \ln{\left(\dfrac{x - 3}{x}\right)} - \ln{(x^2 - 9)} \\[3ex] \ln{\left(\dfrac{x}{x + 3}\right)} + \ln{\left(\dfrac{x - 3}{x}\right)} = \ln{\left[\left(\dfrac{x}{x + 3}\right) * \left(\dfrac{x - 3}{x}\right)\right]}...Law\: 1...Log \\[3ex] \ln{\left(\dfrac{x}{x + 3}\right)} + \ln{\left(\dfrac{x - 3}{x}\right)} = \ln{\left(\dfrac{x - 3}{x + 3}\right)} \\[3ex] x^2 - 9 = x^2 - 3^2 \\[2ex] x^2 - 3^2 = (x + 3)(x - 3) ...Difference\: of\: Two\: Squares \\[2ex] \ln{(x^2 - 9)} = \ln{(x + 3)(x - 3)} \\[2ex] = \ln{\left(\dfrac{x - 3}{x + 3}\right)} - \ln{\left[(x + 3)(x - 3)\right]} \\[3ex] = \ln{\left[\left(\dfrac{x - 3}{x + 3}\right) \div (x + 3)(x - 3)\right]} ...Law\: 2...Log \\[3ex] = \ln{\left[\left(\dfrac{x - 3}{x + 3}\right) * \dfrac{1}{(x + 3)(x - 3)}\right]} \\[3ex] = \ln{\left[\dfrac{1}{(x + 3)^2}\right]} \\[3ex] \dfrac{1}{(x + 3)^2} = (x + 3)^{-2} ...Law\: 6...Exp \\[3ex] = \ln{(x + 3)^{-2}} \\[2ex] = -2 \ln{(x + 3)} ...Law\: 5...Log \\[2ex]$
(16.) $\log{\left[\dfrac{x(x + 3)}{(x + 7)^{12}}\right]}$

$\log{\left[\dfrac{x(x + 3)}{(x + 7)^{12}}\right]} \\[3ex] = \log{x(x + 3)} - \log{(x + 7)^{12}} ...Law\: 2...Log \\[2ex] \log{x(x + 3)} = \log{x} + \log{(x + 3)} ...Law\: 1...Log \\[2ex] \log{(x + 7)^{12}} = 12\log{(x + 7)} ...Law\: 5...Log \\[2ex] = \log{x} + \log{(x + 3)} - 12\log{(x + 7)}$
(17.) $\ln{\left[\dfrac{x^2 - x - 2}{(x + 4)^3}\right]}^{\dfrac{1}{2}}$

$\ln{\left[\dfrac{x^2 - x - 2}{(x + 4)^3}\right]}^{\dfrac{1}{2}} \\[5ex] x^2 - x - 2 = (x + 1)(x - 2) \\[3ex] = \ln{\left[\dfrac{(x + 1)(x - 2)}{(x + 4)^3}\right]}^{\dfrac{1}{2}} \\[5ex] \left[\dfrac{(x + 1)(x - 2)}{(x + 4)^3}\right]^{\dfrac{1}{2}} = \dfrac{(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}}{(x + 4)^{3 * \dfrac{1}{2}}} ...Law\: 5...Exp \\[7ex] \dfrac{(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}}{(x + 4)^{3 * \dfrac{1}{2}}} = \dfrac{(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}}{(x + 4)^{\dfrac{3}{2}}} \\[7ex] = \ln{\left[\dfrac{(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}}{(x + 4)^{\dfrac{3}{2}}}\right]} \\[7ex] = \ln{(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}} - \ln{(x + 4)^{\dfrac{3}{2}}} ...Law\: 2...Log \\[5ex] \ln{[(x + 1)^{\dfrac{1}{2}}(x - 2)^{\dfrac{1}{2}}]} = \ln{(x + 1)^{\dfrac{1}{2}}} + \ln{(x - 2)^{\dfrac{1}{2}}} ...Law\: 1...Log \\[5ex] \ln{(x + 1)^{\dfrac{1}{2}}} = \dfrac{1}{2}\ln{(x + 1)} ...Law\: 5...Log \\[5ex] \ln{(x - 2)^{\dfrac{1}{2}}} = \dfrac{1}{2}\ln{(x - 2)} ...Law\: 5...Log \\[5ex] \ln{(x + 4)^{\dfrac{3}{2}}} = \dfrac{3}{2}\ln{(x + 4)} ...Law\: 5...Log \\[5ex] = \dfrac{1}{2}\ln{(x + 1)} + \dfrac{1}{2}\ln{(x - 2)} - \dfrac{3}{2}\ln{(x + 4)}$
(18.) $\log_3{\left(\dfrac{x^7}{x - 5}\right)}$

$\log_3{\left(\dfrac{x^7}{(x - 5)}\right)} \\[5ex] = \log_3{x^7} - \log_3{(x - 5)} ...Law\: 2...Log \\[3ex] \log_3{x^7} = 7\log_3{x} ...Law\: 5...Log \\[3ex] = 7\log_3{x} - \log_3{(x - 5)}$
(19.) $\dfrac{1}{2}\log_p{c} - \dfrac{2}{5}\log_p{d} - \dfrac{3}{7}\log_p{e}$

$\dfrac{1}{2}\log_p{c} - \dfrac{2}{5}\log_p{d} - \dfrac{3}{7}\log_p{e} \\[3ex] \dfrac{1}{2}\log_p{c} = \log_p{c^{\dfrac{1}{2}}} ...Law\: 5...Log \\[3ex] \dfrac{2}{5}\log_p{d} = \log_p{d^{\dfrac{2}{5}}} ...Law\: 5...Log \\[3ex] \dfrac{3}{7}\log_p{e} = \log_p{e^{\dfrac{3}{7}}} ...Law\: 5...Log \\[3ex] \log_p{c^{\dfrac{1}{2}}} = \log_p{\sqrt{c}} ...Law\: 7...Exp \\[3ex] \log_p{d^{\dfrac{2}{5}}} = \log_p{\sqrt[5]{d^2}} ...Law\: 7...Exp \\[3ex] \log_p{e^{\dfrac{3}{7}}} = \log_p{\sqrt[7]{e^3}} ...Law\: 7...Exp \\[3ex] \dfrac{1}{2}\log_p{c} - \dfrac{2}{5}\log_p{d} = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}}\right)} ...Law\: 2...Log \\[3ex] = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}}\right)} - \dfrac{3}{7}\log_p{e} = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}} \div e^{\dfrac{3}{7}}\right)} ...Law\: 2...Log \\[3ex] = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}} \div e^{\dfrac{3}{7}}\right)} = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}} * \dfrac{1}{e^{\dfrac{3}{7}}}\right)} \\[3ex] = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}} * \dfrac{1}{e^{\dfrac{3}{7}}}\right)} = \log_p{\left(\dfrac{c^{\dfrac{1}{2}}}{d^{\dfrac{2}{5}}e^{\dfrac{3}{7}}}\right)} \\[3ex] OR \\[2ex] = \log_p{\left(\dfrac{\sqrt{x}}{\sqrt[5]{c^2}\sqrt[7]{e^3}}\right)}$
(20.) $3\log_p{c} + 5\log_p{d} - 7\log_p{e}$

$3\log_p{c} + 5\log_p{d} - 7\log_p{e} \\[2ex] 3\log_p{c} = \log_p{c^3} ...Law\: 5...Log \\[2ex] 5\log_p{d} = \log_p{d^5} ...Law\: 5...Log \\[2ex] 7\log_p{e} = \log_p{e^7} ...Law\: 5...Log \\[2ex] = \log_p{c^3} + log_p{d^5} - \log_p{e^7} \\[2ex] \log_p{c^3} + log_p{d^5} = \log_p{c^3d^5} ...Law\: 1...Log \\[2ex] \log_p{c^3d^5} - \log_p{e^7} = \log_p{\left(\dfrac{c^3d^5}{e^7}\right)} ...Law\: 2...Log$

(21.) $\log_p{\sqrt[7]{\dfrac{c^3}{d^7e^{10}}}}$

$\log_p{\sqrt[7]{\dfrac{c^3}{d^7e^{10}}}} \\[7ex] \sqrt[7]{\dfrac{c^3}{d^7e^{10}}} = \left(\dfrac{c^3}{d^7e^{10}}\right)^{\dfrac{1}{7}} ...Law\: 7...Exp \\[7ex] \left(\dfrac{c^3}{d^7e^{10}}\right)^{\dfrac{1}{7}} = \dfrac{c^{3 * \dfrac{1}{7}}}{d^{7 * \dfrac{1}{7}}e^{10 * \dfrac{1}{7}}} ...Law\: 5...Exp \\[7ex] \dfrac{c^{3 * \dfrac{1}{7}}}{d^{7 * \dfrac{1}{7}}e^{10 * \dfrac{1}{7}}} = \dfrac{c^{\dfrac{3}{7}}}{d * e^{\dfrac{10}{7}}} \\[7ex] = \log_p{\left(\dfrac{c^{\dfrac{3}{7}}}{d * e^{\dfrac{10}{7}}}\right)} \\[7ex] = \log_p{c^{\dfrac{3}{7}}} - \log_p{(d * e^{\dfrac{10}{7}})} ...Law\: 2...Log \\[3ex] \log_p{(d * e^{\dfrac{10}{7}})} = \log_p{d} + \log_p{e^{\dfrac{10}{7}}} ...Law\: 1...Log \\[3ex] = \log_p{c^{\dfrac{3}{7}}} - \log_p{d} - \log_p{e^{\dfrac{10}{7}}} \\[3ex] \log_p{c^{\dfrac{3}{7}}} = \dfrac{3}{7}\log_p{c} ...Law\: 5...Log \\[3ex] \log_p{e^{\dfrac{10}{7}}} = \dfrac{10}{7}\log_p{e} ...Law\: 5...Log \\[3ex] = \dfrac{3}{7}\log_p{c} - \log_p{d} - \dfrac{10}{7}\log_p{e}$
(22.) $\log_p{\left(\dfrac{c^3d^7}{e^{10}p^9}\right)}$

$\log_p{\left(\dfrac{c^3d^7}{e^{10}p^9}\right)} \\[3ex] = \log_p{(c^3d^7)} - log_p{(e^{10}p^9)} ...Law\: 2...Log \\[2ex] \log_p{(c^3d^7)} = \log_p{c^3} + \log_p{d^7} ...Law\: 1...Log \\[2ex] \log_p{c^3} = 3\log_p{c} ...Law\: 5...Log \\[2ex] \log_p{d^7} = 7\log_p{d} ...Law\: 5...Log \\[2ex] log_p{(e^{10}p^9)} = \log_p{e^{10}} + \log_p{p^9} ...Law\:1...Log \\[2ex] \log_p{e^{10}} = 10\log_p{e} ...Law\: 5...Log \\[2ex] \log_p{p^9} = 9\log_p{p} ...Law\: 5...Log \\[2ex] \log_p{p} = 1 ...Law\: 4...Log \\[2ex] = 3\log_p{c} + 7\log_p{d} - (10\log_p{e} + 9 * 1) \\[2ex] = 3\log_p{c} + 7\log_p{d} - (10\log_p{e} + 9) \\[2ex] = 3\log_p{c} + 7\log_p{d} - 10\log_p{e} -9$
(23.) $\dfrac{1}{2}\log_p{(c - 1)} + \dfrac{1}{3}\log_p{(d - 2)} - \dfrac{1}{4}\log_p{(e - 3)}$

$\dfrac{1}{2}\log_p{(c - 1)} + \dfrac{1}{3}\log_p{(d - 2)} - \dfrac{1}{4}\log_p{(e - 3)} \\[3ex] \dfrac{1}{2}\log_p{(c - 1)} = \log_p{(c - 1)^{\dfrac{1}{2}}} ...Law\: 5...Log \\[3ex] \dfrac{1}{3}\log_p{(d - 2)} = \log_p{(d - 2)^{\dfrac{1}{3}}} ...Law\: 5...Log \\[3ex] \dfrac{1}{4}\log_p{(e - 3)} = \log_p{(e - 3)^{\dfrac{1}{4}}} ...Law\: 5...Log \\[3ex] \log_p{(c - 1)^{\dfrac{1}{2}}} = \log_p{\sqrt{c - 1}} ...Law\: 7...Exp \\[3ex] \log_p{(d - 2)^{\dfrac{1}{3}}} = \log_p{\sqrt[3]{d - 2}} ...Law\: 7...Exp \\[3ex] \log_p{(e - 3)^{\dfrac{1}{4}}} = \log_p{\sqrt[4]{e - 3}} ...Law\: 7...Exp \\[3ex] \dfrac{1}{2}\log_p{(c - 1)} + \dfrac{1}{3}\log_p{(d - 2)} = \log_p{(c - 1)^{\dfrac{1}{2}}} + \log_p{(d - 2)^{\dfrac{1}{3}}} \\[3ex] = \log_p{[(c - 1)^{\dfrac{1}{2}}(d - 2)^{\dfrac{1}{3}}]} ...Law\: 1...Log \\[3ex] = \log_p{(c - 1)^{\dfrac{1}{2}}} + \log_p{(d - 2)^{\dfrac{1}{3}}} - \log_p{(e - 3)^{\dfrac{1}{4}}} \\[3ex] = \log_p{[(c - 1)^{\dfrac{1}{2}}(d - 2)^{\dfrac{1}{3}}]} - \log_p{(e - 3)^{\dfrac{1}{4}}} \\[3ex] = \log_p{[(c - 1)^{\dfrac{1}{2}}(d - 2)^{\dfrac{1}{3}} \div (e - 3)^{\dfrac{1}{4}}]} ...Law\: 2...Log \\[3ex] = \log_p{\left[(c - 1)^{\dfrac{1}{2}}(d - 2)^{\dfrac{1}{3}} * \dfrac{1}{(e - 3)^{\dfrac{1}{4}}}\right]} \\[5ex] = \log_p{\left[\dfrac{(c - 1)^{\dfrac{1}{2}}(d - 2)^{\dfrac{1}{3}}} {(e - 3)^{\dfrac{1}{4}}}\right]} \\[3ex] OR \\[2ex] = \log_p{\left[\dfrac{\sqrt{(c - 1)}\sqrt[3]{(d - 2)}}{\sqrt[4]{(e - 3)}}\right]}$
(24.) $\ln (x^3\sqrt[7]{(12 - x)^{10}})$

$\ln (x^3\sqrt[7]{(12 - x)^{10}}) \\[2ex] = \ln x^3 + \ln \sqrt[7]{(12 - x)^{10}} ...Law\: 1...Log \\[3ex] \ln x^3 = 3 \ln x ...Law\: 5...Log \\[2ex] \sqrt[7]{(12 - x)^{10}} = (12 - x)^{\dfrac{10}{7}} ...Law\: 7...Exp \\[3ex] \ln (12 - x)^{\dfrac{10}{7}} = \dfrac{10}{7} \ln (12 - x) ...Law\: 5...Log \\[3ex] = 3 \ln x + \dfrac{10}{7} \ln (12 - x)$
(25.) $\ln x^5 - 3\ln \sqrt[3]{x^2}$

$\ln x^5 - 3\ln \sqrt[3]{x^2} \\[5ex] \sqrt[3]{x^2} = x^{\dfrac{2}{3}} ...Law\: 7...Exp \\[5ex] 3\ln \sqrt[3]{x^2} = 3\ln x^{\dfrac{2}{3}} \\[5ex] 3\ln x^{\dfrac{2}{3}} = \ln \left(x^{\dfrac{2}{3}}\right)^3 ...Law\: 5...Log \\[5ex] \left(x^{\dfrac{2}{3}}\right)^3 = \left(x^{\dfrac{2}{3} * 3}\right) ...Law\: 5...Exp \\[5ex] \left(x^{\dfrac{2}{3} * 3}\right) = x^2 \\[5ex] \ln \left(x^{\dfrac{2}{3}}\right)^3 = \ln x^2 \\[5ex] 3\ln \sqrt[3]{x^2} = \ln x^2 \\[5ex] \Rightarrow \ln x^5 - \ln x^2 \\[3ex] = \ln \left(\dfrac{x^5}{x^2}\right) ...Law\: 2...Log \\[5ex] \left(\dfrac{x^5}{x^2}\right) = x^{5 - 2} = x^3 ...Law\: 2...Exp \\[5ex] = \ln x^3$
(26.) $\dfrac{1}{2}\log_p{c} + \dfrac{2}{5}\log_p{d} + \dfrac{3}{7}\log_p{e}$

$\dfrac{1}{2}\log_p{c} + \dfrac{2}{5}\log_p{d} + \dfrac{3}{7}\log_p{e} \\[3ex] \dfrac{1}{2}\log_p{c} = \log_p{c^{\dfrac{1}{2}}} ...Law\: 5...Log \\[3ex] \dfrac{2}{5}\log_p{d} = \log_p{d^{\dfrac{2}{5}}} ...Law\: 5...Log \\[3ex] \dfrac{3}{7}\log_p{e} = \log_p{e^{\dfrac{3}{7}}} ...Law\: 5...Log \\[3ex] \log_p{c^{\dfrac{1}{2}}} = \log_p{\sqrt{c}} ...Law\: 7...Exp \\[3ex] \log_p{d^{\dfrac{2}{5}}} = \log_p{\sqrt[5]{d^2}} ...Law\: 7...Exp \\[3ex] \log_p{e^{\dfrac{3}{7}}} = \log_p{\sqrt[7]{e^3}} ...Law\: 7...Exp \\[3ex] = \log_p{(c^{\dfrac{1}{2}}d^{\dfrac{2}{5}}e^{\dfrac{3}{7}})} ...Law\: 1...Log \\[3ex] OR \\[2ex] = \log_p{(\sqrt{c}\sqrt[5]{d^2}\sqrt[7]{e^3})} ...Law\: 1...Log$
(27.) $\log{\sqrt{c^3d^7}}$

$\log{\sqrt{c^3d^7}} \\[2ex] \sqrt{c^3d^7} = (c^3d^7)^{\dfrac{1}{2}} ...Law\: 7...Exp \\[3ex] (c^3d^7)^{\dfrac{1}{2}} = c^{3 * \dfrac{1}{2}} * d^{7 * \dfrac{1}{2}} ...Law\: 5...Exp \\[3ex] (c^3d^7)^{\dfrac{1}{2}} = c^{\dfrac{3}{2}} * d^{\dfrac{7}{2}} \\[3ex] = \log{(c^{\dfrac{3}{2}} * d^{\dfrac{7}{2}})} \\[3ex] = \log{c^{\dfrac{3}{2}}} + \log{d^{\dfrac{7}{2}}} ...Law\: 1...Log \\[3ex] = \dfrac{3}{2}\log{c} + \dfrac{7}{2}\log{d} ...Law\: 5...Log$
(28.) $\ln {\left(\dfrac{x}{e^x}\right)}$

$\ln {\left(\dfrac{x}{e^x}\right)} \\[5ex] = \ln x - \ln {e^x} ...Law\: 2...Log \\[3ex] \ln {e^x} = x\ln e = 1 ...Law\: 5...Log \\[3ex] \ln e = \log_e{e} = 1 ...Law\: 4...Log \\[3ex] = \ln x - x\ln e \\[3ex] = \ln x - (x * 1) \\[3ex] = \ln x - x$
(29.)

$\log{\sqrt{c^3d^7}} \\[2ex] \sqrt{c^3d^7} = (c^3d^7)^{\dfrac{1}{2}} ...Law\: 7...Exp \\[3ex] (c^3d^7)^{\dfrac{1}{2}} = c^{3 * \dfrac{1}{2}} * d^{7 * \dfrac{1}{2}} ...Law\: 5...Exp \\[3ex] (c^3d^7)^{\dfrac{1}{2}} = c^{\dfrac{3}{2}} * d^{\dfrac{7}{2}} \\[3ex] = \log{(c^{\dfrac{3}{2}} * d^{\dfrac{7}{2}})} \\[3ex] = \log{c^{\dfrac{3}{2}}} + \log{d^{\dfrac{7}{2}}} ...Law\: 1...Log \\[3ex] = \dfrac{3}{2}\log{c} + \dfrac{7}{2}\log{d} ...Law\: 5...Log$
(30.) $\log\left(\sqrt{x^5y^{-8}}\right)$

$\log\left(\sqrt{x^5y^{-8}}\right) \\[3ex] \sqrt{x^5y^{-8}} = (x^5y^{-8})^{\dfrac{1}{2}}...Law\;7...Exp \\[5ex] = \log (x^5y^{-8})^{\dfrac{1}{2}} \\[5ex] = \dfrac{1}{2}\log(x^5y^{-8})...Law\;5...Log \\[5ex] = \dfrac{1}{2}\left(\log x^5 + \log y^{-8}\right)...Law\;1...Log \\[5ex] = \dfrac{1}{2}\left(5\log x + -8\log y\right)...Law\;5...Log \\[5ex] = \dfrac{1}{2}\left(5\log x - 8\log y\right) \\[5ex] = \dfrac{5}{2}\log x - 4\log y$