Change of Base
Convert a logarithm to a different base.
Formula
Variables
Example
log_2(8)=ln8/ln2=3
How it works
The change-of-base formula rewrites a logarithm with one base in terms of logarithms with another base, such as natural logs or common logs. It comes from setting y = log_b x, which means b^y = x, then taking log_k of both sides and using the power rule: y log_k b = log_k x, so y = (log_k x)/(log_k b). This works because all valid logarithm bases measure the same exponent, just on different scales.
When to use it
Use this when your calculator only has ln or log but you need a log with a different base. It is also useful in algebra, calculus, and scientific work when comparing growth rates, solving exponential equations, or simplifying expressions involving unfamiliar log bases.
Worked example, step by step
Find log_5 125 using natural logarithms. Set up the change of base formula: log_5 125 = ln(125) / ln(5) Rewrite 125 as a power of 5: 125 = 5^3 So ln(125) / ln(5) = ln(5^3) / ln(5) Use the power rule: ln(5^3) = 3ln(5) Therefore 3ln(5) / ln(5) = 3 Final answer: log_5 125 = 3
Common mistakes to avoid
- Swapping the numerator and denominator, which gives the reciprocal of the correct value
- Using different bases in the numerator and denominator instead of the same new base k in both places
- Forgetting that the argument x and the base b must both be positive, with b not equal to 1
Did You Know?
The change-of-base formula lets a calculator with only log and ln buttons compute a logarithm in any base at all.
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