Logarithm - Practice Questions

LOGARITHM — Practice Questions

99 multiple-choice questions on logarithms.
Try to solve each question independently before checking the answer key.

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Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.

Quick Notes:
Definition: \({}^{a}\!\log b = c \iff a^c = b\) (with \(a>0, a\neq 1, b>0\))

Logarithm Rules:
\({}^{a}\!\log(xy) = {}^{a}\!\log x + {}^{a}\!\log y\)
\({}^{a}\!\log\dfrac{x}{y} = {}^{a}\!\log x - {}^{a}\!\log y\)
\({}^{a}\!\log x^n = n \cdot {}^{a}\!\log x\)
\({}^{a^n}\!\log x^m = \dfrac{m}{n} \cdot {}^{a}\!\log x\)
\({}^{a}\!\log b = \dfrac{{}^{c}\!\log b}{{}^{c}\!\log a}\) (change of base)
\({}^{a}\!\log b \cdot {}^{b}\!\log c = {}^{a}\!\log c\)
\({}^{a}\!\log b = \dfrac{1}{{}^{b}\!\log a}\)
\(a^{{}^{a}\!\log b} = b\)
\({}^{a}\!\log a = 1\), \(\quad {}^{a}\!\log 1 = 0\)

Problem 1

If \(x\) satisfies the equation \(x \cdot {}^{10}\!\log x = 1000\), then \({}^{100}\!\log x\) equals ...

A. \(-4\) or \(3\) B. \(-3\) or \(3\) C. \(-2\) or \(2\) D. \(-1\) or \(1\) E. \(-\frac{1}{2}\) or \(\frac{1}{2}\)

Problem 2

If \(x\) satisfies the equation \({}^{4}\!\log {}^{4}\!\log x - {}^{4}\!\log {}^{4}\!\log {}^{4}\!\log 16 = 2\), then \({}^{16}\!\log x\) equals ...

A. \(4\) B. \(2\) C. \(1\) D. \(-2\) E. \(-4\)

Problem 3

If \(x_1\) and \(x_2\) satisfy the equation \(\frac{{}^{10}\!\log x^5}{{}^{10}\!\log x} - {}^{10}\!\log x = \frac{5}{{}^{10}\!\log x}\), then \(x_1 + x_2 = \dots\)

A. \(5\) B. \(6\) C. \(60\) D. \(110\) E. \(1100\)

Problem 4

If \(t = \frac{x^2 - 3}{3x + 7}\), then \(\log(1 - |t|)\) can be determined for ...

A. \(2 < x < 6\) B. \(-2 < x < 5\) C. \(-2 \le x \le 6\) D. \(x \le -2\) or \(x > 6\) E. \(x < -1\) or \(x > 3\)

Problem 5

The solutions of the equation \({}^{3}\!\log(9x + 18) = 2 + x\) are \(p\) and \(q\). Then \(p + q = \dots\)

A. \({}^{3}\!\log 3\) B. \({}^{3}\!\log 9\) C. \({}^{3}\!\log 18\) D. \({}^{3}\!\log 216\) E. \({}^{3}\!\log 726\)

Problem 6

If \(\frac{1}{2}\log(2x^2 - x - 2) = \log(x + 2)\), then the maximum value of \(f(y) = -y^2 + 4xy + 5x^2\) equals ...

A. \(302\) B. \(306\) C. \(212\) D. \(318\) E. \(324\)

Problem 7

For \(a > 0\) and \(b > 0\), \({}^{a^m}\!\log b^n = \dots\)

A. \(\frac{m}{n} \cdot {}^{a}\!\log b\) B. \(\frac{n}{m} \cdot {}^{a}\!\log b\) C. \((a \log b)^{m/n}\) D. \(a \log b^m\) E. \(\frac{n}{m} \cdot {}^{b}\!\log a\)

Problem 8

The product of all values of \(x\) that satisfy the equation \(\log\left(64 \cdot 2^{2(x^2 - 40x)}\right) = 0\) is ...

A. \(144\) B. \(100\) C. \(72\) D. \(5\) E. \(6\)

Problem 9

The product of the roots of the equation \({}^{3}\!\log x(2 + {}^{3}\!\log x) = 15\) is ...

A. \(\frac{1}{9}\) B. \(\frac{1}{3}\) C. \(1\) D. \(3\) E. \(9\)

Problem 10

If \(a = {}^{6}\!\log 5\) and \(b = {}^{5}\!\log 4\), then \({}^{4}\!\log 0{,}24 = \dots\)

A. \(\frac{a+2}{ab}\) B. \(\frac{2a+1}{ab}\) C. \(\frac{a-2}{ab}\) D. \(\frac{2a+1}{2ab}\) E. \(\frac{1-2a}{ab}\)

Problem 11

The value of \(x\) that satisfies the equation \({}^{5-4x}\!\log(x^2 - 7x - 5) = \log 10\) is ...

A. \(-4\) B. \(-3\) C. \(-2\) D. \(3\) E. \(2\)

Problem 12

If \(x_1\) and \(x_2\) are the roots of the equation \(x^{(2+\log x)} = 1000\), then \(x_1 \cdot x_2\) equals ...

A. \(10^{-1}\) B. \(10^{-2}\) C. \(10^{0}\) D. \(10\) E. \(100\)

Problem 13

If \(x_1\) and \(x_2\) are the roots of the equation \((\log(x+2))^2 + \log(x+2)^3 = \log 0{,}01\), then \(|x_1 - x_2| = \dots\)

A. \(0{,}9\) B. \(0{,}11\) C. \(0{,}011\) D. \(0{,}09\) E. \(0{,}009\)

Problem 14

\({}^{a}\!\log\left(\frac{1}{b}\right) \cdot {}^{b}\!\log\left(\frac{1}{c}\right) \cdot {}^{c}\!\log\left(\frac{1}{a}\right) = \dots\)

A. \(1 - abc\) B. \(1 + abc\) C. \(\frac{1}{abc}\) D. \(-1\) E. \(1\)

Problem 15

All values of \(x\) that satisfy the inequality \(\frac{1}{2}\log(1 - 2x) < 3\) are ...

A. \(x > \frac{7}{16}\) B. \(x < \frac{7}{16}\) C. \(x < \frac{7}{18}\) D. \(x > \frac{7}{18}\) E. \(x \le \frac{7}{16}\)

Problem 16

If \({}^{9}\!\log 8 = 3m\), then \({}^{4}\!\log 3 = \dots\)

A. \(\frac{1}{4m}\) B. \(\frac{3}{4m}\) C. \(\frac{3}{2m}\) D. \(\frac{m}{4}\) E. \(4m\)

Problem 17

The value of \(x\) that satisfies the equation \(\begin{pmatrix} {}^{x}\!\log y & {}^{2}\!\log z \\ 1 & {}^{3}\!\log y \end{pmatrix} = \begin{pmatrix} {}^{4}\!\log z & 2 \\ 1 & \frac{1}{2} \end{pmatrix}\) is ...

A. \(\sqrt{3}\) B. \(3\) C. \(\sqrt{2}\) D. \(-3\) E. \(0\)

Problem 18

If \((2x)^{1+2\log 2x} > 64x^3\), then ...

A. \(\frac{1}{4} < x < 4\) B. \(x < \frac{1}{4}\) or \(x > 4\) C. \(x < 4\) D. \(0 < x < \frac{1}{4}\) or \(x > 4\) E. \(x > \frac{1}{4}\)

Problem 19

If \({}^{2}\!\log(x + 5) + {}^{2}\!\log(3 - x) < {}^{2}\!\log(4 - x)\), then ...

A. \(x < -\frac{1}{2}(1 + \sqrt{45})\) or \(x > -\frac{1}{2}(1 - \sqrt{45})\) B. \(-5 < x < -\frac{1}{2}(1 + \sqrt{45})\) or \(-\frac{1}{2}(1 - \sqrt{45}) < x < 3\) C. \(-5 < x < -\frac{1}{2}(1 + \sqrt{45})\) or \(-\frac{1}{2}(1 - \sqrt{45}) < x < 4\) D. \(-5 < x < -\frac{1}{2}(1 + \sqrt{45})\) or \(x > 3\) E. \(-5 < x < -\frac{1}{2}(1 + \sqrt{45})\) or \(x > 4\)

Problem 20

The solution set of the inequality \(\log(x + 3) + 2\log 2 > \log x^2\) is ...

A. \(\{x \mid -3 < x < 0\}\) B. \(\{x \mid -2 < x < 0\} \cup \{x \mid 0 < x < 6\}\) C. \(\{x \mid -2 < x < 6\}\) D. \(\{x \mid -3 < x < -2\} \cup \{x \mid 0 < x < 6\}\) E. \(\{x \mid x < -2\} \cup \{x \mid x > 6\}\)

Problem 21

If \(f(x) = \frac{{}^{11}\!\log x}{1 - 2{}^{11}\!\log x}\), then \(f(x) + f\left(\frac{11}{x}\right)\) equals ...

A. \(-11\) B. \(-9\) C. \(-7\) D. \(-2\) E. \(-1\)

Problem 22

Given the system of equations \({}^{5}\!\log x + {}^{5}\!\log y = 5\) and \({}^{5}\!\log x^4 - {}^{5}\!\log y^3 = -1\). The sum of \(x\) and \(y\) that satisfy the system is ...

A. \(225\) B. \(150\) C. \(100\) D. \(75\) E. \(50\)

Problem 23

If \(x_1\) and \(x_2\) are the roots of the equation \(\log(x^2 + 7x + 20) = 1\), then \((x_1 + x_2)^2 - 4x_1x_2\) is ...

A. \(39\) B. \(29\) C. \(20\) D. \(19\) E. \(9\)

Problem 24

The solution set of the inequality \(2\log x \le \log(x + 3) + \log 4\) is ...

A. \(\{x \mid -2 \le x \le 6\}\) B. \(\{x \mid x \ge 6\}\) C. \(\{x \mid 0 < x \le 6\}\) D. \(\{x \mid 0 < x \le 2\}\) E. \(\{x \mid 0 < x < -2 \text{ or } x = 6\}\)

Problem 25

If \({}^{a}\!\log\left(1 - {}^{3}\!\log \frac{1}{27}\right) = 2\), then the value of \(a\) that satisfies is ...

A. \(\frac{1}{8}\) B. \(\frac{1}{4}\) C. \(2\) D. \(3\) E. \(4\)

Problem 26

The values of \(t\) that satisfy \(4 \cdot \left(\frac{1}{2}\log t\right) < \frac{1}{2}\log 81\) are ...

A. \(t > 3\) B. \(-3 < t < 3\) C. \(0 < t < 3\) D. \(-3 < t < 0\) E. \(t < -3\) or \(t > 3\)

Problem 27

If \({}^{4}\!\log(4^x \cdot 4) = 2 - x\), then \(x = \dots\)

A. \(-1\) B. \(-\frac{1}{2}\) C. \(\frac{1}{2}\) D. \(1\) E. \(2\)

Problem 28

The value of \(x\) that satisfies \(\frac{1}{{}^{2}\!\log x} - \frac{1}{{}^{2}\!\log x - 1} < 1\) is ...

A. \(x < 1\) or \(x > 2\) B. \(1 < x < 2\) C. \(0 < x < 2\) D. \(x < 2\) or \(x > 3\) E. \(0 < x < 1\) or \(x > 2\)

Problem 29

If \({}^{2}\!\log a + {}^{2}\!\log b = 12\) and \({}^{3}\!\log a - {}^{2}\!\log b = 4\), then \(a + b = \dots\)

A. \(144\) B. \(272\) C. \(528\) D. \(1024\) E. \(1040\)

Problem 30

If \(b = a^4\), \(a\) and \(b\) positive, then \({}^{a}\!\log b - {}^{b}\!\log a\) is ...

A. \(0\) B. \(1\) C. \(2\) D. \(3\frac{3}{4}\) E. \(4\frac{1}{4}\)

Problem 31

\(\log x = \frac{1}{3}\log 8 + \log 9 - \frac{1}{3}\log 27\) is satisfied for \(x\) equal to ...

A. \(8\) B. \(6\) C. \(4\) D. \(2\) E. \(1\)

Problem 32

If \(\log(y + 2) + 2\log x = 1\), then \(y = \dots\)

A. \(\frac{1}{x^2} - 2\) B. \(\frac{5}{x} - 2\) C. \(\frac{10}{x^2} - 2\) D. \(\frac{1}{2x} - 2\) E. \(8 - x^2\)

Problem 33

If \({}^{9}\!\log 8 = p\), then \({}^{4}\!\log \frac{1}{3}\) equals ...

A. \(-\frac{3}{2p}\) B. \(-\frac{3}{4p}\) C. \(-\frac{2}{3p}\) D. \(\frac{3}{4p}\) E. \(-\frac{6}{4p}\)

Problem 34

The sum of the solutions of the equation \(3{}^{2}\!\log^2 x + 5 \cdot {}^{2}\!\log x + 6 = 0\) is ...

A. \(\frac{1}{4}\) B. \(\frac{3}{4}\) C. \(\frac{1}{8}\) D. \(\frac{3}{8}\) E. \(-\frac{5}{8}\)

Problem 35

If \(2\log y = 3\log(x + 1) + 2\), then ...

A. \(y = x + 3\) B. \(y = 3x + 3\) C. \(y^2 = -9(x + 1)\) D. \(y^2 = 9(x + 1)\) E. \(y^2 = 3(x + 1)\)

Problem 36

If \(2\log x + \log 6x - \log 2x - \log 27 = 0\), then \(x\) equals ...

A. \(3\) B. \(-3\) C. \(3\) or \(-3\) D. \(9\) E. \(9\) or \(-9\)

Problem 37

If \({}^{25}\!\log 5^{2x} = 8\), then \(x = \dots\)

A. \(\frac{1}{4}\) B. \(\frac{1}{2}\) C. \(6\) D. \(8\) E. \(10\)

Problem 38

\({}^{a}\!\log \frac{1}{b} \cdot {}^{b}\!\log \frac{1}{c^2} \cdot {}^{c}\!\log \frac{1}{a^3} = \dots\)

A. \(-6\) B. \(6\) C. \(\frac{b}{a^2c}\) D. \(\frac{a^2c}{b}\) E. \(-\frac{1}{6}\)

Problem 39

If \(2x + y = 8\) and \(\log(x + y) = \frac{3}{2}\log 2 + {}^{8}\!\log 36\), then \(x^2 + 3y = \dots\)

A. \(28\) B. \(22\) C. \(20\) D. \(16\) E. \(12\)

Problem 40

If \({}^{a}\!\log b = 4\), \({}^{c}\!\log a = 2\), and \(a, b, c\) positive, \(a \neq 1, c \neq 1\), then \({}^{a}\!\log(bc)^{\frac{1}{2}} = \dots\)

A. \(2\sqrt{6}\) B. \(3\sqrt{2}\) C. \(16\) D. \(36\) E. \(64\)

Problem 41

The solution of the inequality \(2\log(x + 1) \le \log(x + 4) + \log 4\) is ...

A. \(x \le 7\) B. \(x > 5\) C. \(-1 < x \le 5\) D. \(-1 \le x \le 6\) E. \(x \ge 6\)

Problem 42

The graph of the function \(y = \log x^2\) is ...

A. (graph A) B. (graph B) C. (graph C) D. (graph D) E. (graph E)

Problem 43

If \({}^{a}\!\log 3 = {}^{b}\!\log 27\), \(a > 0, b > 0, a \neq 1, b \neq 1\), then \({}^{a}\!\log b = \dots\)

A. \(\frac{1}{9}\) B. \(\frac{1}{3}\) C. \(1\) D. \(3\) E. \(9\)

Problem 44

The value of \(x\) that satisfies the inequality \({}^{2}\!\log(2x + 7) > 2\) is ...

A. \(x > -\frac{7}{2}\) B. \(-\frac{7}{2} < x < -\frac{3}{2}\) C. \(-\frac{7}{2} < x < \frac{3}{2}\) D. \(x > -\frac{3}{2}\) E. \(-\frac{3}{2} < x < 0\)

Problem 45

Given \(\log 2 = 0{,}3010\) and \(\log 3 = 0{,}4771\), then \(\log(\sqrt{2} \times \sqrt{3}) = \dots\)

A. \(0{,}1505\) B. \(0{,}1590\) C. \(0{,}2007\) D. \(0{,}3389\) E. \(0{,}3891\)

Problem 46

The value of \(x\) that satisfies the inequality \(\frac{1}{\log x} - \frac{1}{2\log x - 1} < 1\) is ...

A. \(0 < x < 1\) B. \(0 < x < \sqrt{10}\) C. \(1 < x < \sqrt{10}\) D. \(0 < x < \sqrt{10}\) or \(x > \sqrt{10}\) E. \(0 < x < 1\) or \(x > \sqrt{10}\)

Problem 47

If \(a, b, c\) are positive numbers with \(b \neq 1\), and \({}^{b}\!\log a = x\), \({}^{b}\!\log c = y\), then \({}^{b}\!\log\frac{\frac{1}{a}\frac{1}{b}c}{\frac{a}{c}b}\) is ...

A. \((a - b)(\frac{x}{b} + \frac{1}{c} + \frac{y}{a})\) B. \((a + b)(\frac{x}{b} + \frac{1}{c} - \frac{y}{a})\) C. \((a - b)(\frac{x}{b} - \frac{1}{c} + \frac{y}{a})\) D. \((a + b)(\frac{x}{b} - \frac{1}{c} + \frac{y}{a})\) E. \((a + b)(\frac{x}{b} - \frac{1}{c} - \frac{y}{a})\)

Problem 48

The value of \(x\) that satisfies the system of equations \(5^{x+y} = 49\) and \(x - y = 6\) is ...

A. \(3 + \frac{1}{2}{}^{5}\!\log 7\) B. \(\frac{1}{2}(3 + {}^{5}\!\log 7)\) C. \(49 + {}^{5}\!\log 7\) D. \(6{}^{5}\!\log 7\) E. \(3 + {}^{5}\!\log 7\)

Problem 49

The values of \(x\) that satisfy \(\frac{1}{2}\log(x^2 - 3) > 0\) are ...

A. \(-\sqrt{3} < x < \sqrt{3}\) B. \(-2 < x < 2\) C. \(-2 < x < -\sqrt{3}\) or \(\sqrt{3} < x < 2\) D. \(x \ge 2\) or \(x \le -2\) E. \(x > 2\) or \(x < \sqrt{3}\)

Problem 50

The values of \(x\) that satisfy \({}^{2}\!\log x - {}^{x}\!\log 2 > 0\) are ...

A. \(x > \frac{1}{2}\) B. \(x > 1\) C. \(\frac{1}{2} < x < 1\) or \(x > 2\) D. \(-1 < x < 0\) or \(x > 1\) E. \(1 < x < 2\)

Problem 51

The value of \(x\) that satisfies the equation \({}^{(3x+2)}\!\log 27 = {}^{5}\!\log 3\) is ...

A. \(42\) B. \(41\) C. \(39\) D. \(7\frac{2}{3}\) E. \(7\frac{1}{3}\)

Problem 52

If \(x_1\) and \(x_2\) satisfy the equation \((2\log x - 1)\frac{1}{x_1 \log 10} = \log 10\), then \(x_1 \cdot x_2 = \dots\)

A. \(5\sqrt{10}\) B. \(4\sqrt{10}\) C. \(3\sqrt{10}\) D. \(2\sqrt{10}\) E. \(\sqrt{10}\)

Problem 53

The value of \(x\) that satisfies \(\log x = 4\log(a+b) + 2\log(a-b) - 3\log(a^2-b^2) - \log\frac{a+b}{a-b}\) is ...

A. \((a+b)\) B. \((a-b)\) C. \((a+b)^2\) D. \(10\) E. \(1\)

Problem 54

The value of \(x\) that satisfies the equation \({}^{2}\!\log{}^{2}\!\log(2^{x+1} + 3) = 1 + {}^{2}\!\log x\) is ...

A. \(\log\frac{2}{3}\) B. \({}^{2}\!\log 3\) C. \({}^{3}\!\log 2\) D. \(-1\) or \(3\) E. \(8\) or \(\frac{1}{2}\)

Problem 55

If \({}^{3}\!\log 5 = p\) and \({}^{5}\!\log 4 = q\), then \({}^{4}\!\log 15 = \dots\)

A. \(\frac{pq}{1+p}\) B. \(\frac{p+q}{pq}\) C. \(\frac{p+1}{pq}\) D. \(\frac{p+1}{q+1}\) E. \(\frac{pq}{1-p}\)

Problem 56

If \(x_1\) and \(x_2\) satisfy the equation \({}^{2}\!\log x(1 + {}^{2}\!\log x) = 2\), then \(x_1 + x_2 = \dots\)

A. \(2\frac{1}{4}\) B. \(2\frac{1}{2}\) C. \(4\frac{1}{4}\) D. \(4\frac{1}{2}\) E. \(6\frac{1}{4}\)

Problem 57

The maximum value of \(f(x) = {}^{4}\!\log(x + 5) + {}^{4}\!\log(3 - x)\) is ...

A. \(2\) B. \(4\) C. \(6\) D. \(8\) E. \(16\)

Problem 58

The sum of the roots of the equation \(\log\frac{x^2+16}{x} = 1\) is ...

A. \(10\) B. \(6\) C. \(2\) D. \(0\) E. \(-2\)

Problem 59

If \({}^{2}\!\log\frac{1}{a} = \frac{3}{2}\) and \({}^{16}\!\log b = 5\), then \({}^{a}\!\log\frac{1}{b^3} = \dots\)

A. \(40\) B. \(-40\) C. \(\frac{40}{3}\) D. \(-\frac{40}{3}\) E. \(20\)

Problem 60

The value of \(x\) that satisfies \(({}^{b}\!\log x)^2 + 10 < 7 \cdot {}^{b}\!\log x\) with \(b > 1\) is ...

A. \(2 < x < 5\) B. \(x < 2\) or \(x > 5\) C. \(b^2 < x < b^5\) D. \(x < b^2\) or \(x > b^5\) E. \(2b < x < 5b\)

Problem 61

If \(m, n, x > 1\), then \(\frac{{}^{n}\!\log x}{1 + {}^{n}\!\log m} = \dots\)

A. \({}^{m+n}\!\log x\) B. \({}^{x}\!\log mn\) C. \({}^{mn}\!\log x\) D. \({}^{x}\!\log(m+n)\) E. \((m+n){}^{mn}\!\log x\)

Problem 62

If \(\frac{{}^{2}\!\log a}{{}^{3}\!\log b} = m\) and \(\frac{{}^{3}\!\log a}{{}^{2}\!\log b} = n\), \(a > 1\) and \(b > 1\), then \(\frac{m}{n} = \dots\)

A. \({}^{2}\!\log 3\) B. \({}^{3}\!\log 2\) C. \({}^{4}\!\log 9\) D. \(({}^{3}\!\log 2)^2\) E. \(({}^{2}\!\log 3)^2\)

Problem 63

If \(x > y > 1\) and \(x^2 + 4y^2 = 12xy\), then \(\log\frac{(x+2y)^2}{(x-2y)^2} = \dots\)

A. \(2\) B. \(4\) C. \(-\log 2\) D. \(\log 2\) E. \(2\log 2\)

Problem 64

If \({}^{2}\!\log x + 2{}^{4}\!\log y = 2\) and \({}^{2}\!\log\frac{x-y}{3} = 0\), then \(x + y = \dots\)

A. \(1\) B. \(3\) C. \(4\) D. \(5\) E. \(6\)

Problem 65

If \({}^{10}\!\log x = b\), then \({}^{10^x}\!\log 100 = \dots\)

A. \(\frac{1}{b+1}\) B. \(\frac{2}{b+1}\) C. \(\frac{1}{b}\) D. \(\frac{2}{b}\) E. \(\frac{2}{10b}\)

Problem 66

If \(a = 0{,}111\dots\), then \({}^{a}\!\log 729 = \dots\)

A. \(-5\) B. \(-4\) C. \(-3\) D. \(4\) E. \(5\)

Problem 67

The inequality \({}^{5}\!\log(x^2 - 2x + 10) < 2\) has solutions for ...

A. \(-5 < x < 3\) B. \(-3 < x < 5\) C. \(x < -5\) or \(x > 3\) D. \(x < -5\) or \(x > 5\) E. \(3 < x < 5\)

Problem 68

The value of \(x\) that satisfies \(\left| \begin{array}{cc} {}^{4}\!\log(x - 3) & {}^{4}\!\log(x - 2) \\ -3 & 2 \end{array} \right| \le 1\) is ...

A. \(3 \le x \le 4\) B. \(1 \le x \le 4\) C. \(3 \le x < 4\) D. \(2 < x \le 3\) E. \(3 < x \le 4\)

Problem 69

If \({}^{2}\!\log\sqrt{x^2 - 16} = 2\), then \({}^{x}\!\log 2 = \dots\)

A. \(\frac{1}{5}\) B. \(\frac{2}{5}\) C. \(\frac{3}{5}\) D. \(\frac{4}{5}\) E. \(4\)

Problem 70

The value of \(x\) that satisfies \(b^{2x} + 10 < 7b^x\) with \(b > 1\) is ...

A. \(x < {}^{b}\!\log 2\) B. \(x > {}^{b}\!\log 5\) C. \(x < {}^{b}\!\log 2\) or \(x > {}^{b}\!\log 5\) D. \({}^{b}\!\log 2 < x < {}^{b}\!\log 5\) E. \(x > {}^{b}\!\log 2\)

Problem 71

If \(a > 1\), \(b > 1\), and \(c > 1\), then \({}^{b}\!\log\sqrt{a} \cdot {}^{c}\!\log b^2 \cdot {}^{a}\!\log\sqrt{c} = \dots\)

A. \(\frac{1}{4}\) B. \(\frac{1}{2}\) C. \(1\) D. \(2\) E. \(3\)

Problem 72

If \({}^{\frac{1}{a}}\!\log \frac{1}{b} = 2\), then ...

A. \({}^{b}\!\log a = 2\) B. \({}^{a}\!\log b = 2\) C. \({}^{\frac{1}{a}}\!\log b = \frac{1}{2}\) D. \({}^{a}\!\log \frac{1}{b} = 2\) E. \({}^{b}\!\log \frac{1}{a} = \frac{1}{2}\)

Problem 73

If \({}^{4}\!\log {}^{4}\!\log x - {}^{4}\!\log {}^{4}\!\log {}^{4}\!\log 16 = 2\), then ...

A. \({}^{2}\!\log x = 8\) B. \({}^{2}\!\log x = 4\) C. \({}^{4}\!\log x = 8\) D. \({}^{4}\!\log x = 16\) E. \({}^{16}\!\log x = 8\)

Problem 74

The value of \(x\) that satisfies the equation \(({}^{4}\!\log x)^2 - {}^{2}\!\log\sqrt{x} - \frac{3}{4} = 0\) is ...

A. \(16\) or \(4\) B. \(16\) or \(\frac{1}{4}\) C. \(8\) or \(2\) D. \(8\) or \(\frac{1}{2}\) E. \(8\) or \(4\)

Problem 75

If \({}^{4}\!\log 6 = m + 1\), then \({}^{9}\!\log 8 = \dots\)

A. \(\frac{3}{4m-2}\) B. \(\frac{3}{4m+2}\) C. \(\frac{3}{2m+4}\) D. \(\frac{3}{2m-4}\) E. \(\frac{3}{2m+2}\)

Problem 76

If \(a > 1\), then the solution of \(({}^{a}\!\log(2x + 1))({}^{3}\!\log\sqrt{a}) = 1\) is ...

A. \(1\) B. \(2\) C. \(3\) D. \(4\) E. \(5\)

Problem 77

If \(x_1\) and \(x_2\) satisfy the equation \((x + 1)^{\log(x+1)} = \frac{(x+1)^3}{100}\), then \(x_1 + x_2 = \dots\)

A. \(81\) B. \(96\) C. \(108\) D. \(120\) E. \(144\)

Problem 78

If \({}^{3}\!\log 4 = a\) and \({}^{3}\!\log 5 = b\), then \({}^{8}\!\log 20 = \dots\)

A. \(\frac{a+b}{2a}\) B. \(\frac{a+b}{3a}\) C. \(\frac{2a+b}{3a}\) D. \(\frac{3a+3b}{3a}\) E. \(\frac{a+2b}{3a}\)

Problem 79

If \({}^{27}\!\log 8 = m\), then \({}^{8}\!\log 144 = \dots\)

A. \(\frac{2}{3m}\) B. \(\frac{3}{2m}\) C. \(\frac{2m}{3(m+1)}\) D. \(\frac{2(2m+1)}{3m}\) E. \(\frac{3(m+1)}{2m}\)

Problem 80

\(\frac{({}^{5}\!\log 10)^2 - ({}^{5}\!\log 2)^2}{{}^{3}\!\log\sqrt{20}} = \dots\)

A. \(\frac{1}{2}\) B. \(1\) C. \(2\) D. \(4\) E. \(5\)

Problem 81

If \(u = x^2\) and \({}^{x}\!\log 10 = {}^{u}\!\log(5u - 40)\), then the value of \(u\) is ...

A. \(25\) B. \(26\) C. \(27\) D. \(28\) E. \(30\)

Problem 82

If \(x_1\) and \(x_2\) with \(x_1 < x_2\) satisfy the equations \({}^{3}\!\log a = 2x^2 + x\), \({}^{9}\!\log b = 5x - x^2\) and \(9a = b\), then \(\frac{x_2}{x_1} = \dots\)

A. \(12\) B. \(10\) C. \(9\) D. \(8\) E. \(6\)

Problem 83

The value of \(x\) that satisfies the equation \(10^{4\log x} - 5(10^{2\log x}) = -4\) is ...

A. \(1\) B. \(4\) C. \(1\) or \(2\) D. \(1\) or \(4\) E. \(2\) or \(4\)

Problem 84

If \({}^{3}\!\log 2 = p\) and \({}^{2}\!\log 7 = q\), then \({}^{14}\!\log 54 = \dots\)

A. \(\frac{p+3}{p+q}\) B. \(\frac{2p}{p+q}\) C. \(\frac{p+3}{p(q+1)}\) D. \(\frac{p+q}{p(q+1)}\) E. \(\frac{p(q+1)}{p+q}\)

Problem 85

If \({}^{4}\!\log 6 = m + 1\), then \({}^{9}\!\log 8 = \dots\)

A. \(\frac{3}{2m+4}\) B. \(\frac{3}{4m+2}\) C. \(\frac{3}{4m-2}\) D. \(\frac{3}{2m-4}\) E. \(\frac{3}{2m+2}\)

Problem 86

If \(f(n) = {}^{2}\!\log 3 \cdot {}^{3}\!\log 4 \cdot {}^{4}\!\log 5 \cdots {}^{n-1}\!\log n\), then \(\sum_{k=2}^{10} f(2^k) = \dots\)

A. \(46\) B. \(48\) C. \(50\) D. \(52\) E. \(54\)

Problem 87

If \(x\) satisfies \({}^{2}\!\log {}^{3}\!\log(x + 2) = 1\) and \(y\) satisfies \(({}^{4}\!\log(3y - 1))({}^{2}\!\log a) = 3\), then the value of \(x + y\) is ...

A. \(2\) B. \(3\) C. \(6\) D. \(9\) E. \(12\)

Problem 88

If \(x_1\) and \(x_2\) are the roots of the equation \((5 - 2\log x)\log x = \log 1000\), then \(x_1^2 + x_2^2 = \dots\)

A. \(0\) B. \(10\) C. \(100\) D. \(1000\) E. \(1100\)

Problem 89

If \(a = {}^{9}\!\log(3\sqrt{16})\) and \(b = {}^{2}\!\log\left(\frac{1}{3}\right)\), then \(ab = \dots\)

A. \(\frac{4}{3}\) B. \(\frac{2}{3}\) C. \(\frac{4}{9}\) D. \(-\frac{2}{3}\) E. \(-\frac{4}{3}\)

Problem 90

\({}^{x}\!\log 2 + {}^{x}\!\log(3x - 4) = 2\) has roots \(x_1\) and \(x_2\) with \(x_1 > x_2\), then \(x_1 - x_2 = \dots\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

Problem 91

The infinite geometric series \((\log(x - 5))^2 + (\log(x - 5))^3 + (\log(x - 5))^4 + \dots\) has a sum for \(x\) that satisfies ...

A. \(-1 < x < 1\) B. \(4 < x < 6\) C. \(5 < x < 6\) D. \(5{,}1 < x < 6\) E. \(5{,}1 < x < 15\)

Problem 92

If \({}^{7}\!\log 2 = a\) and \({}^{2}\!\log 3 = b\), then \({}^{6}\!\log 98 = \dots\)

A. \(\frac{a}{a+b}\) B. \(\frac{a+2}{b+1}\) C. \(\frac{a+2}{a(b+1)}\) D. \(\frac{a+1}{b+2}\) E. \(\frac{a+2}{b(a+1)}\)

Problem 93

If \(\frac{{}^{3}\!\log x}{{}^{3}\!\log w} = 2\) and \({}^{xy}\!\log w = \frac{2}{5}\), then the value of \(\frac{{}^{2}\!\log x}{{}^{2}\!\log y}\) is ...

A. \(8\) B. \(6\) C. \(4\) D. \(2\) E. \(1\)

Problem 94

If \({}^{3}\!\log a + 2({}^{3}\!\log b) = 1\) and \({}^{3}\!\log b + 2({}^{3}\!\log a) = 2\), then the value of \(ab\) is ...

A. \(2\) B. \(3\) C. \(6\) D. \(9\) E. \(12\)

Problem 95

If \({}^{p}\!\log a = 2\) and \({}^{q}\!\log 8p = 2\), then \({}^{2p}\!\log \frac{p^2a}{q} = \dots\)

A. \(3{}^{2}\!\log 2p\) B. \({}^{2}\!\log 2p\) C. \(\frac{3}{{}^{2}\!\log 2p}\) D. \(\frac{1}{{}^{2}\!\log 2p}\) E. \(\frac{3}{{}^{2}\!\log p}\)

Problem 96

If \(x_1\) and \(x_2\) are the solutions of the equation \(({}^{2}\!\log x)^2 + {}^{2}\!\log x = 6\), then \(x_1 x_2 = \dots\)

A. \(2\) B. \(\frac{1}{2}\) C. \(\frac{1}{8}\) D. \(-3\) E. \(-6\)

Problem 97

Given \(f(n) = {}^{3}\!\log 4 \cdot {}^{4}\!\log 5 \cdots {}^{n-1}\!\log n\). If \(a_1\) and \(a_2\) are solutions of the equation \(f(a) + f(a^2) + \dots + f(a^9) = f(a) \cdot f(a^5)\), then \(a_1 \cdot a_2 = \dots\)

A. \(3^7\) B. \(3^8\) C. \(3^9\) D. \(3^{10}\) E. \(3^{11}\)

Problem 98

If \({}^{p^2+4}\!\log 2 = \frac{{}^{3}\!\log 5}{{}^{2}\!\log 5 \cdot {}^{3}\!\log 8}\), with \(p > 0\), then \(p + {}^{p^2}\!\log 16 = \dots\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

Problem 99

Given \({}^{p}\!\log 2 = 9\) and \({}^{q}\!\log 4 = 8\). If \(s = p^3\) and \(t = q^2\), then the value of \({}^{t}\!\log s\) is ...

A. \(\frac{1}{4}\) B. \(\frac{1}{2}\) C. \(\frac{2}{3}\) D. \(\frac{3}{2}\) E. \(2\)