Change of Base Formula

Logarithm Setup

log₅(25) = ?

Base (b)

Argument (x)

Evaluate log_b(x). Base must be positive and ≠ 1; argument must be positive.

Display conversion using

Base 10 (log) Base e (ln) Show Both

Examples

Result

Enter a base and argument above, then press Convert & Evaluate to see the change of base conversion and decimal result.

Value of log_b(x)

—

Base 10 Conversion

—

Natural Log (ln) Conversion

—

Both Methods Agree

—

Exact Integer Answer

—

Change of Base Formula

log_b(x) = log(x) / log(b) = ln(x) / ln(b)

The change of base formula lets you evaluate any logarithm using only the base-10 log or natural log buttons found on every calculator.

The key insight: the ratio of logs is what matters, not which base you use. Both log(x)/log(b) and ln(x)/ln(b) give exactly the same decimal answer — you can verify this in the result panel above.

The formula works for any base b where b > 0 and b ≠ 1, and any argument x > 0.

When an answer is a whole number, the argument is a perfect power of the base — for example log₅(25) = 2 because 5² = 25.

Why It Works

If log_b(x) = y, then b^y = x

Algebraic proof sketch: Let y = log_b(x), which means b^y = x by definition of logarithm.

Take log of both sides: log(b^y) = log(x)

Apply the power rule: y · log(b) = log(x)

Divide both sides by log(b): y = log(x) / log(b)

Since y = log_b(x), we arrive at: log_b(x) = log(x) / log(b)

The same derivation works with ln in place of log, giving ln(x) / ln(b).

Base b must satisfy b > 0 and b ≠ 1.
Argument x must satisfy x > 0.
Negative results are valid — they occur when 0 < x < 1 (for b > 1).
Use any consistent new base — log or ln both work.

Logarithms still confusing?