Logarithms Test — 10 Questions

Question 1

Compute $\log_2 8$.

Accepted Answer

$2^3 = 8$, so $\log_2 8 = 3$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$8$

Question 2

Compute $\log_3 81$.

Accepted Answer

$3^4 = 81$, so $\log_3 81 = 4$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$9$

Question 3

Compute $\log_5 \dfrac{1}{25}$.

Accepted Answer

$\dfrac{1}{25} = 5^{-2}$, so $\log_5 \dfrac{1}{25} = -2$.

Answer

$-2$

Answer

$-1$

Answer

$1$

Answer

$2$

Question 4

Using $\log_a(xy) = \log_a x + \log_a y$, compute $\log_2(4 \cdot 8)$.

Accepted Answer

$\log_2(4 \cdot 8) = \log_2 4 + \log_2 8 = 2 + 3 = 5$. (Check: $4 \cdot 8 = 32 = 2^5$.)

Answer

$3$

Answer

$5$

Answer

$6$

Answer

$32$

Question 5

Compute $\log_3 27 - \log_3 3$.

Accepted Answer

$\log_3 27 = 3$ and $\log_3 3 = 1$, so the difference is $2$. Equivalently, $\log_3 \dfrac{27}{3} = \log_3 9 = 2$.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$9$

Question 6

Compute $\log_2(2^{10})$.

Accepted Answer

By the identity $\log_a(a^k) = k$, $\log_2(2^{10}) = 10$.

Answer

$2$

Answer

$5$

Answer

$10$

Answer

$1024$

Question 7

Solve for $x$: $\log_2 x = 4$.

Accepted Answer

$\log_2 x = 4 \Leftrightarrow x = 2^4 = 16$.

Answer

$2$

Answer

$8$

Answer

$16$

Answer

$32$

Question 8

If $\log_a 5 = 2$, what is $a$? (Assume $a > 0$, $a 
eq 1$.)

Accepted Answer

$\log_a 5 = 2 \Leftrightarrow a^2 = 5 \Leftrightarrow a = \sqrt{5}$ (since $a > 0$).

Answer

$\sqrt{5}$

Answer

$2$

Answer

$5$

Answer

$25$

Question 9

Solve for $x$: $\log_2(x) + \log_2(x-2) = 3$.

Accepted Answer

Combine: $\log_2(x(x-2)) = 3 \Rightarrow x^2 - 2x = 8 \Rightarrow x^2 - 2x - 8 = 0 \Rightarrow (x-4)(x+2) = 0$. Domain requires $x > 2$, so $x = 4$.

Answer

$x = 2$

Answer

$x = 4$

Answer

$x = 6$

Answer

$x = 8$

Question 10

Compute $\log_{10} 1000 + \log_{10} 0.01$.

Accepted Answer

$\log_{10} 1000 = 3$ and $\log_{10} 0.01 = -2$. Sum: $3 + (-2) = 1$. (Or: $\log_{10}(1000 \cdot 0.01) = \log_{10} 10 = 1$.)

Answer

$0$

Answer

$1$

Answer

$2$

Answer

$5$

Logarithms Test — 10 Questions

Logarithm properties — quick reference

Logarithms Test — 10 Questions

Logarithm properties — quick reference