Change of base is one of 3 logarithms formulas in the algebra section of this library, and it is used at high school level.
Why change of base works
Start from b to the y equals x and take the natural log of both sides. The exponent comes out front, giving y times ln b equals ln x, and dividing isolates y. The new base is nothing more than a common measuring stick that both sides are converted into.
What each symbol means
$a$ is any valid new base and $b$ is the original base.
Change of base: when it holds
$x>0$; $a,b>0$ and neither base equals $1$.
When it stops applying
The denominator is the log of the original base, so it must not be zero. That rules out a base of 1, since the log of 1 is 0 in every system, and it also rules out any base that is zero or negative because those logs do not exist.
Change of base: a worked example
$\log_2 10=\ln(10)/\ln(2)\approx3.3219$.
The mistake to avoid
What people do: Computing the log of x divided by b instead of the log of x divided by the log of b.
Why it goes wrong: For log base 2 of 10, the correct value is about 3.3219, while ln of 10 over 2 is ln 5, which is about 1.6094. Dividing inside the log is the quotient rule, a different rule entirely.
Do this instead: Write two separate logarithms with a fraction bar between them, and evaluate each one before dividing.
Change of base: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is any valid new base and $b$ is the original base.
- Check the conditions before substituting. $x>0$; $a,b>0$ and neither base equals $1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Logarithms
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where change of base comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about change of base
Which new base should I choose?
Whichever your calculator has. Base 10 gives 1 divided by 0.30103, and base e gives 2.302585 divided by 0.693147, and both come out to 3.3219 for log base 2 of 10.
Why does my calculator only offer log and ln?
Because those two bases are enough. This formula converts any base into either of them, so the two buttons cover every logarithm you will ever need to evaluate.
Does the conversion work between any two bases?
Yes, in both directions, and there is no restriction beyond each base being positive and not equal to 1. The same ratio pattern applies whichever pair you pick.
Can I use it to show that log base b of a equals 1 over log base a of b?
Yes, just swap the roles and the fraction flips. Since log base 10 of 2 is 0.30103 and its reciprocal is 3.3219, that matches log base 2 of 10 exactly.