Change of Base Formula
Converts a logarithm from one base to another.
Formula
Variables
Example
log₂ 10 = log₁₀ 10 / log₁₀ 2 = 1 / 0.301 ≈ 3.32.
How it works
A logarithm asks, "what exponent on the base gives this number?" The change-of-base formula rewrites that question using any convenient base, such as 10 or e, by dividing two logarithms with the same new base. It works because if y = log_b(x), then b^y = x, and taking logarithms of both sides lets you solve for y in a different base.
When to use it
Use this when your calculator only has log or ln, but the problem gives a different base like 2, 3, or 7. It is also useful for comparing logarithms, simplifying expressions, or solving equations with bases that are not easy to evaluate directly.
Worked example, step by step
Find log_3(20) using common logarithms. log_3(20) = log_10(20) / log_10(3) log_10(20) ≈ 1.3010 log_10(3) ≈ 0.4771 1.3010 / 0.4771 ≈ 2.726 So, log_3(20) ≈ 2.73
Common mistakes to avoid
- Switching the order of the numerator and denominator, which gives the reciprocal instead of the correct value
- Using different bases in the numerator and denominator, such as log_10(x) over ln(b)
- Forgetting that b must be a valid log base, so b > 0 and b ≠ 1
Did You Know?
This trick lets a calculator with only log and ln buttons compute a logarithm in any base at all.
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