Exercises: Understand the Inverse Relationship Between Exponents and Logarithms

$g(x) = x^3$

$g(x) = \log_3(x)$

$g(x) = \dfrac{1}{3^x}$

$g(x) = 3^{-x}$

4

5

6

8

The $x$-axis

The $y$-axis

The line $y = x$

The line $y = 2x$

$\log_{125}(5) = 3$

$\log_5(125) = 3$

$\log_3(125) = 5$

$\log_5(3) = 125$

$\log_{10}(7)$

70

7

$10 \cdot \log_{10}(7)$

$5 \times 125 = 625$

3

25

$\dfrac{125}{5} = 25$

Alex's reasoning is valid — $\log_8(2) = 3$ follows from $2^3 = 8$.

Alex confused the base and argument. The relationship $2^3 = 8$ means $\log_2(8) = 3$, not $\log_8(2) = 3$.

Alex used the wrong exponent — the answer should be $\log_8(2) = \dfrac{1}{3}$, but the notation $\log_8(2)$ is correct.

Alex's notation is fine; the only error is that $2^3 \neq 8$.

$\log_5\!\left(\dfrac{1}{5}\right)$

$\log_5(1)$

$\log_5(0)$

$\log_5(5)$

Sam added instead of multiplied — the correct calculation is $2 + 8 = 10$.

Sam treated the logarithm as multiplication. $\log_2(8)$ asks "what power of 2 equals 8?" — it is not $2 \times 8$.

Sam used the wrong base — the problem should be $\log_8(2)$, not $\log_2(8)$.

Sam's method is correct; the answer is 16.

Correct — splitting $\log_2(4 + 4)$ into $\log_2(4) + \log_2(4)$ follows the sum rule for logarithms.

Incorrect — there is no rule that says $\log_b(x + y) = \log_b(x) + \log_b(y)$. The correct answer is $\log_2(8) = 3$.

Incorrect — Jamie computed $\log_2(4) = 2$ correctly, but the sum should be $2 \times 2 = 4$, not $2 + 2 = 4$.

Incorrect — Jamie should have used $\ln$ instead of $\log_2$ for sum expressions.