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Exponents and Logarithms

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For the Classic ACT exam:
The ACT Mathematics test is a timed exam...60 questions in 60 minutes
This implies that you have to solve each question in one minute.
Each of the first 20 questions (less challenging) will typically take less than a minute a solve.
Each of the next 20 questions (medium challenging) may take about a minute to solve.
Each of the last 20 questions (more challenging) may take more than a minute to solve.
The goal is to maximize your time.
You use the time saved on the questions you solve in less than a minute to solve 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 a wrong answer.
Also: please note that unless specified otherwise, geometric figures are drawn to scale. So, you can figure out the correct answer by eliminating the incorrect options.
Other suggestions are listed in the solutions/explanations as applicable.

These are the solutions to the ACT past questions on the topics: Exponents and Logarithms.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided for some questions.
The link to the video solutions will be provided for you. Please subscribe to the YouTube channel to be notified of upcoming livestreams. You are welcome to ask questions during the video livestreams.
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These are the notable notes regarding factoring

Factoring Formulas

$ \underline{Difference\;\;of\;\;Two\;\;Squares} \\[3ex] (1.)\;\;x^2 - y^2 = (x + y)(x - y) \\[5ex] \underline{Difference\;\;of\;\;Two\;\;Cubes} \\[3ex] (2.)\;\; x^3 - y^3 = (x - y)(x^2 + xy + y^2) \\[5ex] \underline{Sum\;\;of\;\;Two\;\;Cubes} \\[3ex] (3.)\;\; x^3 + y^3 = (x + y)(x^2 - xy + y^2) \\[5ex] $ These are the laws of exponents and the laws of logarithms.

Laws of Exponents and Laws of Logarithms

  • Law 1: Exponents

  • (1.) Product Rule

    $ p^c * p^d = p^{c + d} \\[4ex] p^{c + d} = p^c * p^d $

  • Law 1: Logarithms

  • (1.)

    $ \log_p{c} + \log_p{d} = \log_p{cd} \\[4ex] \log_p{cd} = \log_p{c} + \log_p{d} $

  • Law 2: Exponents

  • (2.) Quotient Rule

    $ p^c \div p^d = p^{c - d} \\[4ex] \dfrac{p^c}{p^d} = p^{c - d} \\[4ex] p^{c - d} = p^c \div p^d \\[4ex] p^{c - d} = \dfrac{p^c}{p^d} $

  • Law 2: Logarithms

  • (2.)

    $ \log_p{c} - \log_p{d} = \log_p({c \div d}) \\[4ex] \log_p{c} - \log_p{d} = \log_p{\left(\dfrac{c}{d}\right)} \\[5ex] \log_p({c \div d}) = \log_p{c} - \log_p{d} \\[4ex] \log_p{\left(\dfrac{c}{d}\right)} = \log_p{c} - \log_p{d} $

  • Law 3: Exponents

  • (3.)

    $ {any\: base}^0 = 1 \\[5ex] p^0 = 1 $

  • Law 3: Logarithms

  • (3.)

    $ \log_{any\: base}{1} = 0 \\[5ex] \log_p{1} = 0 \\[4ex] \ln{1} = \log_e{1} = 0 $

  • Law 4: Exponents

  • (4.)

    $ {any\: base}^1 = any\: base \\[5ex] p^1 = p $

  • Law 4: Logarithms

  • (4.)

    $ \log_{any\: base}{any\: base} = 1 \\[5ex] \log_p{p} = 1 $

  • Law 5: Exponents

  • (5.) Expanded Power Rule

    $ (p)^c = (p^1)^c = p^{1 * c} = p^c \\[5ex] \left(\dfrac{p}{q}\right)^c = \dfrac{p^c}{q^c} \\[7ex] (p^c)^d = p^{c * d} \\[5ex] p^{c * d} = (p^c)^d \\[7ex] \left(\dfrac{p^c}{q^d}\right)^e = \dfrac{p^{ce}}{q^{de}} \\[7ex] (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}} \\[7ex] p^{\dfrac{cd}{e}} = (p^c)^{\dfrac{d}{e}} $

  • Law 5: Logarithms

  • (5.)

    $ \log_p{c^d} = d * \log_p{c} \\[4ex] d * \log_p{c} = \log_p{c^d} $

  • Law 6: Exponents

  • (6.) Rule of Negative Exponents

    $ p^{-c} = \dfrac{1}{p^c} \\[5ex] \dfrac{1}{p^c} = p^{-c} \\[5ex] p^c = \dfrac{1}{p^{-c}} \\[5ex] \dfrac{1}{p^{-c}} = p^c \\[5ex] p^{-\dfrac{c}{d}} = \dfrac{1}{p^{\dfrac{c}{d}}} \\[7ex] \dfrac{1}{p^{\dfrac{c}{d}}} = p^{-\dfrac{c}{d}} \\[7ex] p^{\dfrac{c}{d}} = \dfrac{1}{p^{-\dfrac{c}{d}}} \\[7ex] \dfrac{1}{p^{-\dfrac{c}{d}}} = p^{\dfrac{c}{d}} $

  • Law 6: Logarithms

  • (6.)
    Change of Base of Log

    $ \log_p{d} = \dfrac{\log_c{d}}{\log_c{p}} \\[4ex] \dfrac{\log_c{d}}{\log_c{p}} = \log_p{d} \\[4ex] \log_p{d} * \log_c{p} = \log_c{d} \\[4ex] \log_c{d} = \log_p{d} * \log_c{p} $

  • Law 7: Exponents

  • (7.) Rule of Fractional Exponents

    $ p^{\dfrac{1}{c}} = \sqrt[c]{p} \\[5ex] p^{\dfrac{c}{d}} = \sqrt[d]{p^c} \\[5ex] p^{\dfrac{c}{d}} = (\sqrt[d]{p})^c \\[5ex] \sqrt[d]{p^c} = p^{\dfrac{c}{d}} \\[5ex] (\sqrt[d]{p})^c = p^{\dfrac{c}{d}} $

  • Law 7: Logarithms

  • (7.)

    $ p^{\log_p{c}} = c \\[5ex] c = p^{\log_p{c}} \\[5ex] p^{d\log_p{c}} = p^{\log_p{c^d}} = c^d \\[5ex] c^d = p^{\log_p{c^d}} = p^{d\log_p{c}} \\[5ex] e^{\ln{c}} = c \\[5ex] c = e^{\ln{c}} \\[5ex] e^{d\ln{c}} = e^{\ln{c^d}} = c^d \\[5ex] c^d = e^{\ln{c^d}} = e^{d\ln{c}} $

(1.) Which of the following is equivalent to $\left(x^4\right)^{\dfrac{5}{4}}\left(x^5\right)^{\dfrac{2}{5}}$?

$ A.\;\; x^3 \\[4ex] B.\;\; x^{\dfrac{63}{20}} \\[6ex] C.\;\; x^{\dfrac{9}{2}} \\[6ex] D.\;\; x^7 \\[4ex] E.\;\; x^{10} \\[4ex] $

$ \left(x^4\right)^{\dfrac{5}{4}}\left(x^5\right)^{\dfrac{2}{5}} \\[7ex] x^{4 * \dfrac{5}{4}} \cdot x^{5 * \dfrac{2}{5}}...Law\;5...Exp \\[7ex] x^5 \cdot x^2 \\[4ex] x^{5 + 2} ...Law\;1...Exp \\[4ex] x^7 $
(2.) The expression $\left(x^{10}\right)^4$ is equivalent to:

$ F.\;\; x^{14} \\[4ex] G.\;\; x^{40} \\[4ex] H.\;\; x^{10,000} \\[4ex] J.\;\; 4x^6 \\[4ex] K.\;\; 4x^9 \\[4ex] $

$ \left(x^{10}\right)^4 \\[4ex] = x^{10 * 4} ...Law\;5...Exp \\[4ex] = x^{40} $
(3.) $\log_5{25} =?$

$ A.\;\; \dfrac{1}{5} \\[5ex] B.\;\; 2 \\[3ex] C.\;\; 5 \\[3ex] D.\;\; 25 \\[3ex] E.\;\; 5^{25} \\[4ex] $

$ \log_5{25} \\[4ex] = \log_5{5^2} \\[5ex] = 2\log_5{5} ...Law\;5...Log \\[4ex] = 2 \cdot 1 ...Law\;4...Log \\[3ex] = 2 $

Number 3
(4.) What is the value of $\log_5 625$

$ A.\;\; 3 \\[3ex] B.\;\; 4 \\[3ex] C.\;\; 6 \\[3ex] D.\;\; 125 \\[3ex] E.\;\; 436 \\[3ex] $

$ \log_5{625} \\[4ex] = \log_5{5^4} \\[4ex] = 4 \log_5{5} ...Law\;5\;Log \\[4ex] = 4 \cdot 1 ...Law\;4\;Log \\[4ex] 4 $
(5.) What is the value of x in the equation $\log_3{54} - \log_3{2} = \log_2{x}$

$ F.\;\; 3 \\[3ex] G.\;\; 8 \\[3ex] H.\;\; 9 \\[3ex] J.\;\; 52 \\[3ex] K.\;\; 108 \\[3ex] $

$ \log_3{54} - \log_3{2} = \log_2{x} \\[4ex] \log_3{54 \div 2} = \log_2{x} ...Law\;2...Log \\[4ex] \log_3{27} = \log_2{x} \\[4ex] \log_2{x} = \log_3{3^3} \\[5ex] \log_2{x} = 3\log_3{3} ...Law\;5...Log \\[4ex] \log_2{x} = 3(1) ...Law\;4...Log \\[4ex] \log_2{x} = 3 \\[4ex] x = 2^3...\text{Relationship between Exponents and Logarithms} \\[4ex] x = 8 \\[3ex] $

Calculator 5
(6.) For all negative values of k, what is the range of values of $2^k$

F. All negative numbers
G. All numbers less than 1
H. All rational numbers less than 1
J. All positive numbers less than 1
K. All positive numbers less than 2


Negative values of k implies that k is less than 0
Let us test some numbers for k

$ If\;\;k = -1 \\[3ex] 2^{-1} = \dfrac{1}{2^1} = \dfrac{1}{2} ...Law\;6...Exp \\[5ex] \dfrac{1}{2} \;\;\text{is a rational number} \\[5ex] \dfrac{1}{2} \;\;\text{is less than 1} \\[7ex] If\;\;k = -2 \\[3ex] 2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4} ...Law\;6...Exp \\[5ex] \dfrac{1}{4} \;\;\text{is a rational number} \\[5ex] \dfrac{1}{4} \;\;\text{is less than 1} \\[7ex] If\;\;k = -\dfrac{1}{2} \\[5ex] 2^{-\dfrac{1}{2}} = \dfrac{1}{2^{\dfrac{1}{2}}} ...Law\;6...Exp \\[7ex] \dfrac{1}{2^{\dfrac{1}{2}}} = \dfrac{1}{\sqrt{2}} ...Law\;7...Exp \\[7ex] \dfrac{1}{\sqrt{2}} \;\;\text{is an irrational number} \\[5ex] \dfrac{1}{\sqrt{2}} \;\;\text{is less than 1} \\[5ex] $ This implies that for all negative values of k, the range of values of $2^k$ is all positive numbers less than 1.
(7.)

(8.) For all nonzero values of a, the expression $\dfrac{a^2a^4}{a^6}$ is equal to:

$ F.\;\; 0 \\[3ex] G.\;\; 1 \\[3ex] H.\;\; a \\[3ex] J.\;\; a^2 \\[4ex] K.\;\; a^{12} \\[4ex] $

$ \dfrac{a^2a^4}{a^6} \\[6ex] a^{2 + 4 - 6}...Laws\;1\;\;and\;\;2...Exp \\[6ex] a^0 ...Law\;3...Exp \\[3ex] 1 $
(9.)

(10.) Which of the following expressions is equivalent to $\left(2x\right)^6\left(10y^2\right)$?

$ F.\:\: 20x^6y^2 \\[4ex] G.\:\: 200x^6y^2 \\[4ex] H.\:\: 240x^6y^2 \\[4ex] J.\;\; 640x^6y^2 \\[4ex] K.\:\: 6,400x^6y^2 \\[4ex] $

$ \left(2x\right)^6\left(10y^2\right)...Law\;5...Exp \\[4ex] 2^6 \cdot x^6 \cdot 10 \cdot y^2 \\[4ex] 64 \cdot 10 \cdot x^6 \cdot y^2 \\[4ex] 640x^6y^2 $
(11.) Let n be a positive integer.
Which of the following expressions is equivalent to $0.0002^n$

$ F.\;\; 2^{-2n} \\[4ex] G.\;\; 2^{-3n} \\[4ex] H.\;\; 2^n \times 10^{-n} \\[4ex] J.\;\; 2^n \times 10^{-3n} \\[4ex] K.\;\; 2^n \times 10^{-4n} \\[4ex] $

$ 0.0002^n \\[4ex] = \dfrac{0\color{darkblue}{.0002}^n}{\color{darkblue}{10000}^n} \\[6ex] = \dfrac{2^n}{(10^4)^n} \\[6ex] = \dfrac{2^n}{10^{4n}}...Law\;5...Exp \\[6ex] = 2^n \times 10^{-4n} ...Law\;6...Exp $

Calculator 11-1st
Calculator 11-2nd
(12.) For all nonzero values of x and y, which of the following expressions is equal to $\dfrac{9x^3y^2}{3y} \cdot \dfrac{xy^2}{2x^4}$

$ A.\;\; \dfrac{3y^3}{2} \\[6ex] B.\;\; \dfrac{3y^3}{2x} \\[6ex] C.\;\; \dfrac{9y^3}{5} \\[6ex] D.\;\; \dfrac{6x^6}{y} \\[6ex] E.\;\; 6x^{11} \\[4ex] $

$ \dfrac{9x^3y^2}{3y} \cdot \dfrac{xy^2}{2x^4} \\[6ex] DISSOCIATE \\[3ex] \dfrac{9 \cdot x^3 \cdot y^2 \cdot x \cdot y^2}{3 \cdot y \cdot 2 \cdot x^4} \\[7ex] SOLVE \\[3ex] \dfrac{3}{2} \cdot x^{3 + 1 - 4} \cdot y^{2 + 2 - 1} ...Laws\;1\;\;and\;\;2...Exp \\[6ex] \dfrac{3}{2} \cdot x^0 \cdot y^3 \\[6ex] \dfrac{3}{2} \cdot 1 \cdot y^3...Law\;3...Exp \\[6ex] ASSOCIATE \\[3ex] \dfrac{3y^3}{2} $
(13.)

(14.) The function $f(x) = 5^{\dfrac{x}{2}}$ has an inverse function, $f^{-1}(x)$, defined for all x > 0 by which of the following expressions?

$ A.\;\; \dfrac{2}{\log_5{x}} \\[6ex] B.\;\; \dfrac{1}{\left(\log_5{x}\right)^2} \\[7ex] C.\;\; \left(\log_5{x}\right)^2 \\[5ex] D.\;\; \dfrac{1}{2}\log_5{x} \\[5ex] E.\;\; 2\log_5{x} \\[4ex] $

$ f(x) = 5^{\dfrac{x}{2}} \\[5ex] y = 5^{\dfrac{x}{2}} \\[5ex] \text{Interchange x and y} \\[3ex] x = 5^{\dfrac{y}{2}} \\[5ex] \text{Solve for y} \\[3ex] \text{Introduce Log to both sides} \\[3ex] \log x = \log 5^{\dfrac{y}{2}} \\[5ex] \log 5^{\dfrac{y}{2}} = \log x...Law\;5...Log \\[5ex] \dfrac{y}{2}\log 5 = \log x \\[5ex] \dfrac{y}{2} = \dfrac{\log x}{\log 5}...Law\;6...Log \\[5ex] \dfrac{y}{2} = \log_5{x} \\[5ex] y = 2\log_5{x} $
(15.)

(16.) For all positive real numbers a, $\left(\sqrt[3]{a^{36}}\right)^{\dfrac{1}{2}}$ = ?

$ A.\;\; a^3 \\[3ex] B.\;\; a^6 \\[3ex] C.\;\; a^{15} \\[3ex] D.\;\; a^{30} \\[3ex] E.\;\; a^{54} \\[3ex] $

$ \left(\sqrt[3]{a^{36}}\right)^{\dfrac{1}{2}} \\[5ex] a^{36 * \dfrac{1}{3} * \dfrac{1}{2}} ...Laws\;7\;\;and\;\;5...Exp \\[5ex] a^6 $
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