Logarithms
What Is a Logarithm?
A logarithm answers an exponent question. log base b of x is the exponent y that makes b^y = x, where b is positive, b is not 1, and x is positive.
Reviewed
Exponential and logarithmic forms
The equations b^y = x and log_b(x) = y say exactly the same thing. Switching between them is the most direct way to evaluate basic logarithms.
For example, log_2(8)=3 because 2^3=8. A logarithm does not divide x by the base; it reports the power needed to produce x.
log_b(x)=y if and only if b^y=xDomain and base restrictions
For real logarithms, x must be positive. A positive real base cannot produce zero or a negative number through real exponentiation.
The base must be positive and cannot equal 1. Powers of 1 never produce values other than 1, so they cannot define a one-to-one logarithmic inverse.
x > 0, b > 0, and b != 1The inverse relationship
An exponential function b^x and its logarithm log_b(x) undo each other on their domains. Their graphs are reflections across the line y=x.
This inverse property is why logarithms isolate exponents in equations.
log_b(b^x)=x; b^(log_b x)=x for x>0Worked examples
Evaluate log_3(81)
Ask which power of 3 equals 81.
3^1=33^2=93^3=273^4=81
Result: log_3(81)=4
Solve 10^x=0.001
Write the decimal as a power of ten.
0.001=10^-3The exponents must match
Result: x=-3, so log_10(0.001)=-3
Where this idea is used
- Solving equations where the unknown appears in an exponent.
- Compressing wide ranges in pH, sound, and magnitude scales.
- Turning multiplication into addition for analysis and computation.
Common mistakes
- Treating log_b(x) as x/b.
- Taking a real logarithm of zero or a negative number.
- Using base 1, which cannot produce an inverse function.
Questions about logarithms
Can a logarithm be negative?
Yes. If 0<x<1 and b>1, the exponent needed to produce x is negative.
What does log without a base mean?
Context determines it. Many school calculators use base 10, while higher mathematics and programming often use the natural base e.
Why is log(0) undefined?
No finite real exponent of a positive base equals zero.