Math Core

Since logarithms are exponents, the rules for exponents turn into rules for logarithms. These properties let you break a complicated log into simple pieces, combine several logs into one, and compute a logarithm in any base using only the $\log$ or $\ln$ key on your calculator.

Where the properties come from

Recall the exponent rules $b^m \cdot b^n = b^{m+n}$ , $\dfrac{b^m}{b^n} = b^{m-n}$ and $(b^m)^n = b^{mn}$ . Suppose $\log_b M = m$ and $\log_b N = n$ , which means $M = b^m$ and $N = b^n$ . Then

MN = b^m \cdot b^n = b^{m+n}, \quad\text{so}\quad \log_b(MN) = m + n = \log_b M + \log_b N.

In words: multiplying the inputs adds the logs. The other two rules come from the same kind of argument.

Properties of logarithms

For positive numbers $M$ and $N$ , a base $b > 0$ with $b \ne 1$ , and any real number $p$ :

Property	Rule
Product	$\log_b(MN) = \log_b M + \log_b N$
Quotient	$\log_b \dfrac{M}{N} = \log_b M - \log_b N$
Power	$\log_b\left(M^p\right) = p \log_b M$

For example, $\log_2 8 + \log_2 4 = \log_2 32$ checks out as $3 + 2 = 5$ , and $\log_3 81^{2} = 2\log_3 81 = 2 \cdot 4 = 8$ .

Expanding a logarithm

To expand a log, apply the properties to write it as a sum and difference of simpler logs, with no products, quotients or powers inside. Work from the outside in: first split the fraction, then split products, then bring down exponents.

Worked example: Expand

Expand $\log_3 \dfrac{9x^2}{y}$ .

Solution.

\begin{aligned} \log_3 \frac{9x^2}{y} &= \log_3(9x^2) - \log_3 y && \text{quotient} \\ &= \log_3 9 + \log_3 x^2 - \log_3 y && \text{product} \\ &= 2 + 2\log_3 x - \log_3 y && \text{power, and } \log_3 9 = 2 \end{aligned}

Roots are powers too: $\sqrt{x} = x^{1/2}$ , so $\ln\sqrt{x} = \tfrac{1}{2}\ln x$ .

Condensing logarithms

To condense, run the properties backward to write an expression as a single logarithm. First move every coefficient up into an exponent (power rule). Then combine: added logs multiply, subtracted logs divide.

Worked example: Condense

Write $2\ln x + \ln(x + 1) - 3\ln y$ as a single logarithm.

Solution.

\begin{aligned} 2\ln x + \ln(x + 1) - 3\ln y &= \ln x^2 + \ln(x + 1) - \ln y^3 && \text{power} \\ &= \ln\left(x^2(x + 1)\right) - \ln y^3 && \text{product} \\ &= \ln \frac{x^2(x + 1)}{y^3} && \text{quotient} \end{aligned}

Common mistake

There is no rule for the log of a sum or a difference. $\log(x + y)$ is not $\log x + \log y$ , and $\ln(x + 1)$ can't be split at all. Likewise, $\dfrac{\log M}{\log N}$ is not $\log \dfrac{M}{N}$ . The properties apply only to products, quotients and powers inside a single log.

Using known values

If you know the logs of a few numbers, the properties give you logs of their products, quotients and powers.

Worked example: Build from known logs

Given $\log_7 2 \approx 0.356$ and $\log_7 3 \approx 0.565$ , estimate $\log_7 12$ and $\log_7 1.5$ .

Solution. Write each number using $2$ 's and $3$ 's. Since $12 = 2^2 \cdot 3$ ,

\log_7 12 = 2\log_7 2 + \log_7 3 \approx 2(0.356) + 0.565 = 1.277.

Since $1.5 = \dfrac{3}{2}$ ,

\log_7 1.5 = \log_7 3 - \log_7 2 \approx 0.565 - 0.356 = 0.209.

Change of base

Calculators only have keys for base $10$ and base $e$ . To evaluate a log in any other base, use this formula.

Change-of-base formula

\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}

Why it works: let $y = \log_b x$ , so $b^y = x$ . Take $\ln$ of both sides and use the power rule: $y \ln b = \ln x$ . Divide by $\ln b$ to get $y = \dfrac{\ln x}{\ln b}$ . The same steps work with $\log$ .

Worked example: Change of base

Evaluate $\log_3 20$ to three decimal places.

Solution.

\log_3 20 = \frac{\ln 20}{\ln 3} \approx \frac{2.995732}{1.098612} \approx 2.727.

Check: $3^{2.727} \approx 20$ . It makes sense that the answer is between $2$ and $3$ , since $3^2 = 9$ and $3^3 = 27$ .

Tip

Before you compute a log, estimate it with nearby powers. $\log_3 20$ must be between $2$ and $3$ . If your calculator says $0.367$ , you divided in the wrong order.

Practice

Practice 1

Evaluate $\log_6 4 + \log_6 9$ without a calculator.

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

Practice 2

Evaluate $\log_2 80 - \log_2 5$ without a calculator.

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

Practice 3

Which is the expanded form of $\log \dfrac{x^3 y}{z^2}$ ?

Practice 4

Which single logarithm equals $3\log_2 x - \dfrac{1}{2}\log_2 y + \log_2 5$ ?

Practice 5

If $\log_b x = 3$ and $\log_b y = -2$ , find $\log_b \dfrac{x^2}{y^3}$ .

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

Practice 6

Given $\log 2 \approx 0.3010$ and $\log 3 \approx 0.4771$ , estimate $\log 18$ using properties of logarithms. Give $4$ decimal places.

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

Practice 7

Use the change-of-base formula to evaluate $\log_6 50$ . Round to $3$ decimal places.

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

Practice 8

Find the exact value of $\log_4 8$ .

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