Logarithm Rules: Product, Quotient, Power, and Change of Base

What a logarithm means

A logarithm is another way to write an exponent question.

The expression \( \log_b a \) means: what power should go on \( b \) to get \( a \)?

\[ \log_b a = x \Longleftrightarrow b^x = a \]

For example, \( \log_2 8 = 3 \) because \( 2^3 = 8 \). The logarithm gives the exponent, not the base and not the final value.

Rules before you start

Logarithms have restrictions. In \( \log_b a \), the base must be positive, the base cannot be 1, and the input must be positive.

\[ b > 0,\quad b \ne 1,\quad a > 0 \]

These restrictions are not decoration. If a problem has \( \log_b(x-4) \), then the inside must satisfy \( x - 4 > 0 \), so \( x > 4 \).

Logarithm rules chart

Use this chart as a quick reference, then read the same rules in math notation. The letters \( M \) and \( N \) stand for positive expressions.

Product rule:

\[ \log_b(MN) = \log_b M + \log_b N \]

Quotient rule:

\[ \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \]

Power rule:

\[ \log_b(M^r) = r\log_b M \]

Root rule:

\[ \log_b\sqrt[n]{M} = \frac{1}{n}\log_b M \]

Change-of-base rule:

\[ \log_b M = \frac{\log_c M}{\log_c b} \]

Why the product rule works

The product rule comes from adding exponents with the same base.

Suppose \( \log_b M = p \) and \( \log_b N = q \). That means \( b^p = M \) and \( b^q = N \).

Multiply the two equations:

\[ MN = b^p \cdot b^q = b^{p+q} \]

So the exponent that produces \( MN \) is \( p + q \), which gives:

\[ \log_b(MN) = \log_b M + \log_b N \]

Example 1: combine logarithms

Combine this expression into one logarithm:

\[ \log_2 8 + \log_2 4 \]

The logs have the same base, so use the product rule:

\[ \log_2 8 + \log_2 4 = \log_2(8\cdot4) \]

\[ \log_2(32) = 5 \]

The final answer is \( 5 \) because \( 2^5 = 32 \).

Example 2: expand a logarithm

Expand the expression:

\[ \log_3\left(\frac{27x^2}{y}\right) \]

Use the quotient rule first:

\[ \log_3(27x^2) - \log_3 y \]

Then use the product rule on \( 27x^2 \):

\[ \log_3 27 + \log_3(x^2) - \log_3 y \]

Now simplify and use the power rule:

\[ 3 + 2\log_3 x - \log_3 y \]

This expanded form is valid when \( x > 0 \) and \( y > 0 \).

Example 3: condense an expression

Condense this expression into one logarithm:

\[ 2\log_5 x - \log_5 3 \]

Move the coefficient into the exponent:

\[ 2\log_5 x = \log_5(x^2) \]

Now use the quotient rule:

\[ \log_5(x^2) - \log_5 3 = \log_5\left(\frac{x^2}{3}\right) \]

So the condensed expression is \( \log_5\left(\frac{x^2}{3}\right) \), with \( x > 0 \).

Inverse rules for logs and exponents

Logarithms and exponents undo each other when the base matches.

\[ \log_b(b^x) = x \]

\[ b^{\log_b x} = x \]

These rules are useful when solving equations. They let you move between logarithmic form and exponential form without changing the meaning.

Example 4: solve a logarithmic equation

Solve this equation:

\[ \log_5(x - 1) = 2 \]

Rewrite the equation in exponential form:

\[ x - 1 = 5^2 \]

\[ x - 1 = 25 \]

\[ x = 26 \]

Check the restriction: \( x - 1 > 0 \). Since \( 26 - 1 = 25 \), the solution is allowed.

Change of base

The change-of-base rule lets you evaluate a logarithm even when your calculator only has \( \log \) or \( \ln \).

\[ \log_b M = \frac{\log M}{\log b} = \frac{\ln M}{\ln b} \]

For example:

\[ \log_2 10 = \frac{\log 10}{\log 2} \]

This is especially helpful when the base is not 10 and not \( e \).

Common mistakes

Do not split a sum inside a log. In general, \( \log_b(M + N) \ne \log_b M + \log_b N \). The product rule works for multiplication, not addition.

Do not ignore the base. The expression \( \log_2 8 \) is not the same as \( \log_4 8 \).

Do not apply log rules when the bases are different. You cannot combine \( \log_2 x + \log_3 y \) with the product rule.

Do not forget restrictions. A final answer must make every logarithm input positive.

Quick practice

1. Evaluate \( \log_3 81 \). Answer: \( 4 \).

2. Expand \( \log_2(8x) \). Answer: \( 3 + \log_2 x \), with \( x > 0 \).

3. Condense \( \log_7 a + \log_7 b \). Answer: \( \log_7(ab) \).

4. Condense \( 3\log_4 x \). Answer: \( \log_4(x^3) \), with \( x > 0 \).

5. Solve \( \log_2(x + 5) = 4 \). Answer: \( x = 11 \).

6. Rewrite \( \log_6 14 \) using common logs. Answer: \( \frac{\log 14}{\log 6} \).