Logarithms
Logarithm Rules Explained
Logarithm rules translate exponent laws: products become sums, quotients become differences, and powers become coefficients.
Reviewed
Product and quotient rules
Because multiplying equal-base powers adds exponents, the logarithm of a product is the sum of logarithms. Dividing powers subtracts exponents, producing the quotient rule.
Each logarithm created by an expansion must have a positive argument in real-number work.
log_b(MN)=log_b(M)+log_b(N); log_b(M/N)=log_b(M)-log_b(N)Power rule
An exponent on a positive logarithm argument becomes a multiplier. This rule is especially useful for isolating an unknown exponent.
Be careful with even powers: log(x^2)=2log(x) is not a valid real identity for negative x because log(x) is undefined. A domain-aware version is log(x^2)=2log(|x|) for x not equal to zero.
log_b(M^p)=p log_b(M), for M>0Change of base
To evaluate a logarithm whose base is not available on a calculator, divide two logarithms taken in any common valid base.
The numerator measures the target value and the denominator rescales that measurement to one unit of the desired base.
log_b(x)=ln(x)/ln(b)=log_10(x)/log_10(b)Common mistakes
- Splitting a sum: log(M+N) is not log(M)+log(N).
- Expanding into logarithms whose individual arguments are not positive.
- Dividing by log(x) instead of log(base) in the change-of-base formula.
Worked examples
Expand log_2(8x^3/y)
Assume x and y are positive.
log_2(8)+log_2(x^3)-log_2(y)3+3log_2(x)-log_2(y)
Result: 3 + 3log_2(x) - log_2(y)
Evaluate log_5(17)
Use natural logarithms for change of base.
log_5(17)=ln(17)/ln(5)2.833213/1.609438
Result: log_5(17) is approximately 1.76037
Where this idea is used
- Solving exponential equations.
- Simplifying products and powers before numerical evaluation.
- Computing arbitrary-base logarithms with ln or log10 keys.
Questions about logarithm rules
Is there a rule for log(M+N)?
No general rule separates a logarithm of a sum.
Can the logarithm base be between 0 and 1?
Yes, as long as it is positive and not 1. The resulting logarithmic function is decreasing.
Why must the change-of-base denominator be nonzero?
log_c(b)=0 only when b=1, which is already excluded as a logarithm base.