Samuel Dominic Chukwuemeka (SamDom For Peace)

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 Chukwuemeka

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.

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 $


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(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 $