Change of Base Formula Formula

The Formula

\log_b x = \frac{\log_a x}{\log_a b}
Most commonly: \log_b x = \frac{\ln x}{\ln b} or \log_b x = \frac{\log_{10} x}{\log_{10} b}.

When to use: Your calculator only has \ln and \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_5 20 = \frac{\ln 20}{\ln 5} = \frac{2.996}{1.609} \approx 1.861
Check: 5^{1.861} \approx 20. \checkmark

Notation

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

What This Formula Means

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

Your calculator only has \ln and \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_b x = \frac{\log_a x}{\log_a b} for any valid base a; in particular \log_b x = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Worked Examples

Example 1

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

Solution

  1. 1
    The change of base formula: \log_b(a) = \frac{\ln(a)}{\ln(b)} (or equivalently \frac{\log(a)}{\log(b)}).
  2. 2
    Apply: \log_5(20) = \frac{\ln(20)}{\ln(5)}.
  3. 3
    Calculate: \frac{\ln(20)}{\ln(5)} = \frac{2.9957}{1.6094} \approx 1.861.

Answer

\log_5(20) \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 e). This is essential because most calculators only have \log and \ln buttons.

Example 2

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

Common Mistakes

  • Flipping the fraction: \log_b x = \frac{\ln x}{\ln b}, NOT \frac{\ln b}{\ln x}. The base goes in the denominator.
  • Using the formula when it's not needed: if the problem gives \log_2 8, just think 2^? = 8, so the answer is 3. No formula required.
  • Forgetting that \log_b b = 1 for any base b—this is a quick sanity check for the change-of-base formula.

Why This Formula Matters

Essential for calculator computation, comparing logarithmic scales, and proving that all logarithmic functions have the same basic shape (just vertically scaled).

Frequently Asked Questions

What is the Change of Base Formula formula?

A formula for converting a logarithm from one base to another: \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 and \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 a. The two most common choices are a = e (using \ln) and a = 10 (using \log).

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

Essential for calculator computation, comparing logarithmic scales, and proving that all logarithmic functions have the same basic shape (just vertically scaled).

What do students get wrong about Change of Base Formula?

The argument x goes in the numerator, the base b goes in the denominator. A mnemonic: 'argument on top, base on bottom.'

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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