The Logarithmic Function

The logarithmic function is the inverse of the exponential function. One takes an exponent and returns a result; the other takes a result and returns the exponent. Everything about the graph of $\log_b(x)$ — where it crosses the axes, which way it runs — follows from that single relationship.

Concept The two functions side by side

Exponential:

f(x) = b^x

— you know the exponent, you want the result.

Logarithmic:

f(x) = \log_b(x)

— you know the result, you want the exponent.

Each undoes the other, so their graphs are mirror images reflected in the line $y = x$ .

Concept Where each graph crosses an axis

The exponential crosses the

y

-axis at

(0,\ 1)

, because

b^0 = 1

for any base.

The logarithmic crosses the

x

-axis at

(1,\ 0)

, because

\log_b(1) = 0

for any base.

The two points are reflections of each other, which is exactly what you would expect from inverse functions. In the exponential, $x$ is the exponent — any base to the power 0 gives 1. In the logarithm, $x$ is the result — when the result is 1, the exponent must be 0.

Concept How the base changes the graph

When

b > 1

, as in

\log_2(x)

, the function is increasing.

When

0 < b < 1

, the function is decreasing.

The restrictions on the base are the same ones that apply to any logarithm:

b > 0 \quad \text{and} \quad b \neq 1

A base of 1 would give only the value 1, so no exponent could ever produce anything else — which is why it is excluded.

Example Reading a point off each graph

Take base 2. Find the exponential value at $x = 3$ , and the logarithm at $x = 8$ .

f(3) = 2^3 = 8

, so the exponential passes through

(3,\ 8)

\log_2(8) = 3

, so the logarithm passes through

(8,\ 3)

The coordinates are swapped — the mark of inverse functions.

⟹ (3, 8) and (8, 3)

Note Mistakes to avoid

Expecting the logarithmic graph to cross the

y

-axis — it never does, since

\log_b(0)

is undefined.

Swapping the two intercepts: the exponential meets

(0,1)

, the logarithm meets

(1,0)

Allowing a base of 1, or a negative base.

Assuming a logarithmic function always increases; a base below 1 makes it decrease.

Forgetting that the input of a logarithm must be positive.

Summary

The logarithmic function is the inverse of the exponential: one finds the result, the other finds the exponent.
Their graphs are reflections of each other in the line y = x.
The exponential passes through (0, 1); the logarithm passes through (1, 0).
A base greater than 1 gives an increasing graph; a base between 0 and 1 gives a decreasing one.
The base must always satisfy b > 0 and b ≠ 1.