Change of Base Formula Formula

Change of base formula is a formula for converting a logarithm from one base to another: _b x = x/ b = x/ b.

The Formula

log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}
Most commonly: log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b} or log⁡bx=log⁡10xlog⁡10b\log_b x = \frac{\log_{10} x}{\log_{10} b}.

When to use: Your calculator only has ln⁡\ln and log⁡10\log_{10} buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each other—changing base just changes the scale factor.

Quick Example

log⁡520=ln⁡20ln⁡5=2.9961.609≈1.861\log_5 20 = \frac{\ln 20}{\ln 5} = \frac{2.996}{1.609} \approx 1.861
Check: 51.861≈205^{1.861} \approx 20. \checkmark

Notation

The formula works with any intermediate base aa. The two most common choices are a=ea = e (using ln⁡\ln) and a=10a = 10 (using log⁡\log).

What This Formula Means

A formula for converting a logarithm from one base to another: log⁡bx=ln⁡xln⁡b=log⁡xlog⁡b\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}.

Your calculator only has ln⁡\ln and log⁡10\log_{10} buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each other—changing base just changes the scale factor.

Formal View

log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b} for any valid base aa; in particular log⁡bx=ln⁡xln⁡b=log⁡10xlog⁡10b\log_b x = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Worked Examples

Example 1

easy
Evaluate log⁡5(20)\log_5(20) using the change of base formula and a calculator.

Answer

log⁡5(20)≈1.861\log_5(20) \approx 1.861

First step

1
The change of base formula: log⁡b(a)=ln⁡(a)ln⁡(b)\log_b(a) = \frac{\ln(a)}{\ln(b)} (or equivalently log⁡(a)log⁡(b)\frac{\log(a)}{\log(b)}).

Full solution

  1. 2
    Apply: log⁡5(20)=ln⁡(20)ln⁡(5)\log_5(20) = \frac{\ln(20)}{\ln(5)}.
  2. 3
    Calculate: ln⁡(20)ln⁡(5)=2.99571.6094≈1.861\frac{\ln(20)}{\ln(5)} = \frac{2.9957}{1.6094} \approx 1.861.
The change of base formula converts a logarithm from any base to a quotient of logarithms in a base your calculator supports (usually base 10 or base ee). This is essential because most calculators only have log⁡\log and ln⁡\ln buttons.

Example 2

medium
Show that log⁡4(8)=32\log_4(8) = \frac{3}{2} using the change of base formula.

Example 3

medium
Use change-of-base to evaluate log⁡255\log_{25} 5.

Common Mistakes

  • Flipping the ratio - it is log⁡xlog⁡b\frac{\log x}{\log b} (target over base), not base over target.
  • Switching intermediate bases mid-problem - use the same base aa in numerator and denominator.
  • Thinking the new base must be 10 or e - any base works; those two are just the ones on the calculator.

Why This Formula Matters

Calculators only carry ln⁡\ln and log⁡10\log_{10}, so this formula is the bridge to every other base — essential for solving exponential equations numerically and for graphing arbitrary-base logs. Forgetting it leaves a student stuck staring at log⁡27\log_2 7 with no button to press. Recognizing it by "Is the base something other than ee or 10 that I must turn into a computable ratio of logs?" — rather than by familiar numbers — is what lets a student tell it apart from logarithm properties and natural logarithm and power rule for logs in a mixed problem set.

Frequently Asked Questions

What is the Change of Base Formula formula?

A formula for converting a logarithm from one base to another: log⁡bx=ln⁡xln⁡b=log⁡xlog⁡b\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}.

How do you use the Change of Base Formula formula?

Your calculator only has ln⁡\ln and log⁡10\log_{10} buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each other—changing base just changes the scale factor.

What do the symbols mean in the Change of Base Formula formula?

The formula works with any intermediate base aa. The two most common choices are a=ea = e (using ln⁡\ln) and a=10a = 10 (using log⁡\log).

Why is the Change of Base Formula formula important in Math?

Calculators only carry ln⁡\ln and log⁡10\log_{10}, so this formula is the bridge to every other base — essential for solving exponential equations numerically and for graphing arbitrary-base logs. Forgetting it leaves a student stuck staring at log⁡27\log_2 7 with no button to press. Recognizing it by "Is the base something other than ee or 10 that I must turn into a computable ratio of logs?" — rather than by familiar numbers — is what lets a student tell it apart from logarithm properties and natural logarithm and power rule for logs in a mixed problem set.

What do students get wrong about Change of Base Formula?

The procedure for change of base formula is the easy part; the trap is flipping the ratio. Asking "Is the base something other than ee or 10 that I must turn into a computable ratio of logs?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Change of Base Formula formula?

Before studying the Change of Base Formula formula, you should understand: logarithm, natural logarithm.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Exponents and Logarithms: Rules, Proofs, and Applications →
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