Change of Base Formula — Definition, Formula & Examples

Change of Base Formula

A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.

Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.

Change of base formula:

Example 1:

Example 2: (note that )

Example 3:

Key Formula

\log_a x = \frac{\log_b x}{\log_b a}

Where:

$x$ = The argument of the logarithm (must be positive)
$a$ = The original base of the logarithm (must be positive and not equal to 1)
$b$ = The new base you are converting to (must be positive and not equal to 1)

Worked Example

Problem: Evaluate log₅ 125 using the change of base formula with common logarithms (base 10).

Step 1: Apply the change of base formula, converting from base 5 to base 10.

\log_5 125 = \frac{\log_{10} 125}{\log_{10} 5}

Step 2: Evaluate the numerator. Since 125 = 10^{2.0969...}, use a calculator to find log₁₀ 125.

\log_{10} 125 \approx 2.09691

Step 3: Evaluate the denominator. Use a calculator to find log₁₀ 5.

\log_{10} 5 \approx 0.69897

Step 4: Divide the two results.

\frac{2.09691}{0.69897} = 3

Answer: log₅ 125 = 3, which makes sense because 5³ = 125.

Another Example

This example uses natural logarithms (ln) instead of common logarithms, and the answer is not a whole number. It shows the more typical real-world scenario where the result is an irrational decimal.

Problem: Evaluate log₃ 20 using the change of base formula with natural logarithms (base e). Round to four decimal places.

Step 1: Apply the change of base formula, choosing natural logarithms (ln) as the new base.

\log_3 20 = \frac{\ln 20}{\ln 3}

Step 2: Use a calculator to evaluate ln 20.

\ln 20 \approx 2.99573

Step 3: Use a calculator to evaluate ln 3.

\ln 3 \approx 1.09861

Step 4: Divide to get the final value.

\frac{2.99573}{1.09861} \approx 2.7268

Answer: log₃ 20 ≈ 2.7268

Frequently Asked Questions

Why do you need the change of base formula?

Most calculators only have buttons for log base 10 (log) and log base e (ln). If you need to evaluate a logarithm with any other base — such as log₂, log₃, or log₇ — the change of base formula lets you convert it into an expression involving log or ln that your calculator can handle.

Does it matter which new base you choose?

No, the formula works with any valid base b (positive and not equal to 1). You will get the same final answer whether you use base 10, base e, or any other base. Most students choose base 10 or base e because those are available on a calculator.

Can you use the change of base formula to convert between log and ln?

Yes. For example, you can write log₁₀ x = ln x / ln 10. This means any common logarithm can be expressed in terms of natural logarithms, and vice versa. The conversion factor ln 10 ≈ 2.302585 appears frequently in science and engineering for this reason.

Change of Base Formula vs. Logarithm Power Rule

	Change of Base Formula	Logarithm Power Rule
Formula	log_a(x) = log_b(x) / log_b(a)	log_a(x^n) = n · log_a(x)
Purpose	Converts a logarithm from one base to another	Brings an exponent out of the argument as a coefficient
When to use	When you need to evaluate or simplify a log with an inconvenient base	When the argument of a logarithm is raised to a power
Changes the base?	Yes — that is the entire point	No — the base stays the same

Why It Matters

The change of base formula appears throughout Algebra 2 and Precalculus whenever you solve exponential or logarithmic equations with uncommon bases. It is also essential on standardized tests and in courses like chemistry and computer science, where bases like 2 and e arise naturally. Without it, you would have no practical way to compute logarithms like log₂ 50 or log₇ 300 by hand or with a standard calculator.

Common Mistakes

Mistake: Putting the logs in the wrong order — writing log_b(a) / log_b(x) instead of log_b(x) / log_b(a).

Correction: Remember: the log of the argument (x) goes in the numerator, and the log of the original base (a) goes in the denominator. A helpful mnemonic: the argument stays on top, just as it sits 'on top' of the base in log_a(x) notation.

Mistake: Using two different bases in the numerator and denominator, such as writing ln(x) / log₁₀(a).

Correction: Both the numerator and the denominator must use the same new base b. If you choose ln for one, you must use ln for the other.

Related Terms

Logarithm — The core concept this formula applies to
Common Logarithm — Base-10 log, the most common new base choice
Natural Logarithm — Base-e log, another common new base choice
Logarithm Rules — Product, quotient, and power rules used alongside
e — The base of natural logarithms (≈ 2.71828)
Formula — General term for mathematical relationships like this one