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)The product rule comes from adding exponents
Let x = b^m and y = b^n, with positive x and y. Multiplying the values adds their exponents.
xy = b^m b^n = b^(m+n)Use the product rule for exponents.log_b(xy) = m + nTake log base b of both sides.log_b(xy) = log_b(x) + log_b(y)Replace m and n by their logarithmic definitions.
The quotient and power rules follow from subtraction and multiplication of exponents in the same way.
There is no sum rule for logarithms
Incorrect: log(2 + 3) = log(2) + log(3)
log(5) is not log(6). The right side equals log(2 x 3) = log(6).
Log rules translate multiplication, division, and powers. Addition inside a logarithm has no corresponding single rule.
Worked example 1
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)
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>0Worked example 2
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
Change 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)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.