The Graph of a Logarithmic Function
Simple Explanation
y = log_b(x) is the inverse of the exponential function bˣ — it answers the question "b to what power gives x?" It is only defined for positive x, can output any real number, and has a vertical asymptote at x=0.
Why Do We Need It?
Logarithms "undo" exponential growth, letting you solve for an unknown exponent — essential whenever an exponential relationship needs to be reversed.
See It
A curve defined only for positive x, rising slowly, passing through (1,0), with a vertical asymptote along the y-axis
Formula
Properties of Logarithmic Functions
y = log_b(x) (b>0, b≠1): domain (0,∞), range all reals, x-intercept (1,0), asymptote x=0
A logarithmic function is the inverse of the exponential function with the same base — it is only defined for positive inputs, and can output any real number.
- b
- — the base of the logarithm, a positive number not equal to 1
When to use it: Whenever identifying or sketching the basic logarithmic graph and its key features.
Worked Example
Evaluate a logarithmic function
Evaluate y=log₂(x) at x=8, x=1, and x=1/4.
Why Does This Work?
Since exponential and logarithmic functions are inverses of each other, the logarithmic graph is exactly the exponential graph reflected across the line y=x — every point (a,b) on y=bˣ becomes the point (b,a) on y=log_b(x).
Real-Life Example
Measuring earthquake intensity (the Richter scale)
Earthquake magnitudes are measured on a logarithmic scale, where each whole number increase represents a tenfold increase in shaking amplitude.
Logarithms compress an enormous range of physical intensities into a manageable, human-readable scale — exactly what the Richter scale does.
Practice
Evaluate log₃(9).
EasyCommon mistake
Attempting to evaluate the logarithm of zero or a negative number — logarithms are only defined for positive inputs.
Quick Review
- y=log_b(x): domain (0,∞), range all reals.
- Passes through (1,0); vertical asymptote at x=0.
- The inverse of the exponential function with the same base.