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=xRead a logarithm as a question about an exponent
Each logarithmic statement has an equivalent exponential statement. Switching between them is often the fastest way to evaluate a log exactly.
| Exponential statement | Logarithmic statement | Question being answered |
|---|---|---|
| 2^5 = 32 | log_2(32) = 5 | What power of 2 gives 32? |
| 10^-3 = 0.001 | log_10(0.001) = -3 | What power of 10 gives 0.001? |
| e^0 = 1 | ln(1) = 0 | What power of e gives 1? |
For real logarithms, the input must be positive, the base must be positive, and the base cannot equal 1.
Worked example 1
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
Domain 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 example 2
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
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.