Math Core

Exponent and logarithm problems on the AMC reward fluency: rewriting everything in a common base, spotting a hidden quadratic, and knowing the log rules well enough to chain them. A logarithm is just an exponent, so every log problem is really an exponent problem in disguise.

Logs are exponents

Definition

Logarithm

For $b > 0$ , $b \ne 1$ and $x > 0$ , $\log_b x$ is the exponent you put on $b$ to get $x$ :

\log_b x = y \quad\Longleftrightarrow\quad b^y = x.

So $\log_2 32 = 5$ and $\log_9 27 = \frac{3}{2}$ (since $9^{3/2} = 27$ ). Because $b^{\log_b x} = x$ , a log and an exponential with the same base undo each other.

The graphs of $y = 2^x$ and $y = \log_2 x$ are reflections over $y = x$ . Notice that $\log_2 x$ only exists for $x > 0$ .

y = 2^x and y = log₂ x are inverse functions.Open in grapher →

The rules

Each log rule is an exponent rule read backwards.

Rule	Why
$\log_b (xy) = \log_b x + \log_b y$	$b^m b^n = b^{m + n}$
$\log_b \dfrac{x}{y} = \log_b x - \log_b y$	$\dfrac{b^m}{b^n} = b^{m - n}$
$\log_b x^k = k \log_b x$	$(b^m)^k = b^{km}$
$\log_b x = \dfrac{\log_c x}{\log_c b}$	change of base

Change of base gives three consequences that appear constantly on the AMC:

Change-of-base shortcuts

\log_a b \cdot \log_b c = \log_a c, \qquad \log_a b = \frac{1}{\log_b a}, \qquad \log_{a^m} b^n = \frac{n}{m}\log_a b.

The first one makes long products of logs telescope. The last one lets you rewrite every log in a problem with the same base.

Worked example: A telescoping product

Evaluate $\log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{63} 64$ .

Each base cancels with the previous argument: the whole product is $\log_2 64 = 6$ .

Worked example: Common base

Solve $\log_2 x + \log_4 x + \log_{16} x = 7$ .

Rewrite in base $2$ : $\log_4 x = \frac{1}{2}\log_2 x$ and $\log_{16} x = \frac{1}{4}\log_2 x$ . So

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x = 7,

giving $\log_2 x = 4$ and $x = 16$ .

Hidden quadratics

An equation with $4^x$ and $2^x$ is a quadratic in $t = 2^x$ , since $4^x = (2^x)^2$ . After solving for $t$ , remember that $t = 2^x$ must be positive.

Worked example: A quadratic in disguise

Find the sum of all real solutions of $4^x - 3 \cdot 2^{x + 1} + 8 = 0$ .

Let $t = 2^x$ . Then $4^x = t^2$ and $2^{x + 1} = 2t$ , so $t^2 - 6t + 8 = 0$ and $t = 2$ or $t = 4$ . That gives $x = 1$ or $x = 2$ , with sum $3$ .

Worked example: Checking the domain

Solve $\log_6 x + \log_6 (x - 5) = 2$ .

Combine: $\log_6 \big(x(x - 5)\big) = 2$ , so $x^2 - 5x = 36$ , or $(x - 9)(x + 4) = 0$ . But $x = -4$ makes $\log_6 x$ undefined, so the only solution is $x = 9$ .

Common mistake

Combining logs can create extraneous solutions. The equation $\log(x(x - 5)) = 2$ allows $x = -4$ , but the original equation doesn't. Always plug solutions back into the original logs.

Tip

To count the digits of a huge power $N$ , use $\text{digits} = \lfloor \log_{10} N \rfloor + 1$ . With $\log_{10} 2 \approx 0.30103$ , the number $2^{100}$ has $\lfloor 30.103 \rfloor + 1 = 31$ digits.

Practice

Practice 1

Solve $8^x = 32^{x - 3}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is $\log_3 81 - \log_2 \dfrac{1}{8} + \log_5 \sqrt{5}$ ?

Practice 3

If $\log_a b = 3$ , what is $\log_b \left(a^2 b\right)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Solve $\log_2\big(\log_3(\log_4 x)\big) = 0$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Positive numbers $a, b, c$ satisfy $2^a = 3$ , $3^b = 5$ and $5^c = 8$ . What is $abc$ ?

Practice 6

What is the product of all positive real solutions of $x^{\log_2 x} = 16x^3$ ?

Practice 7

Given that $\log_{10} 2 \approx 0.30103$ , how many digits does $2^{100}$ have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Positive real numbers $x$ and $y$ with $x \lt y$ satisfy $\log_x y + \log_y x = \dfrac{10}{3}$ and $xy = 144$ . What is $x + y$ ?

Real contest practice

2021 AMC 12A, Problem 14: sums of logs with power bases; use $\log_{a^m} b^n = \frac{n}{m}\log_a b$ .
2020 AMC 12A, Problem 10: nested logs with different bases; convert to a common base.
2022 AMC 12A, Problem 14: simplifying a product of log expressions with the log rules.