Inverses of Exponential Functions

One of the central ideas in AP Precalculus is that logarithmic functions are the inverses of exponential functions. This inverse relationship allows us to:

• undo exponential growth and decay

• solve exponential equations

• interpret models involving time, rate, and magnitude

• connect algebraic manipulation to graphical behavior

Understanding logarithms as inverses provides a unified view of these two function families.

Review of Inverse Functions

Two functions $f$ and $g$ are inverses if:

$$f(g(x))=x$$and$$g(f(x))=x$$

Graphically:

• Their graphs are reflections across the line y=x

• The domain of one is the range of the other

Exponential Functions

An exponential function has the form:

$$f(x)=a^x (a>0 , a\neq1)$$

Key properties:

• Domain: (−∞,∞)

• Range: (0,∞)

• Horizontal asymptote: y=0

• One-to-one (passes the horizontal line test)

Because exponential functions are one-to-one, they have inverses.

Defining the Logarithmic Function

The inverse of $$f(x)=b^x$$ is the logarithmic function:

$$f^{-1}(x)=\log_{a}x$$

This definition comes directly from rewriting the exponential equation:

$$a^y=x \longleftrightarrow y=\log_{a}x$$

Thus, a logarithm expresses the exponent needed to obtain a given value.

Inverse Relationship Algebraically

Consider the compositions:

$$f(f^{-1}(x))$$

$$a^{\log_{a}x}=x$$

$$f^{-1}(f(x))$$

$$\log_{a}(a^x)=x$$

These identities confirm that logarithmic and exponential functions are inverses.

Domain and Range Reversal

This reversal is a hallmark of inverse functions and is frequently tested conceptually on the AP exam.

Graphical Interpretation

Reflection Across y=x

• The graph of $$y=\log_{a}x$$ is the mirror image of $$y=a^x$$

Key Features of $$y=\log_{a}x$$

• Vertical asymptote: x=0

• x-intercept: (1,0)

• Increasing if a>1

• Decreasing if 0<a<1

Using Inverses to Solve Equations

Solving Exponential Equations

$$2^{x}=5$$

Apply logarithms:

$$x=\log_{2}5$$

Solving Logarithmic Equations

$$\log_{3}x=4$$

Rewrite exponentially:

$$x=3^4=81$$

Common Errors

• Forgetting that logarithms require positive inputs

• Confusing $$\log_{a}x$$ with $$a^x$$

• Ignoring domain restrictions after inversion

• Treating inverse notation $$f^{-1}(x)$$ as a reciprocal

AP Exam Focus

Students should be able to:

• explain why logarithms are inverses of exponentials

• rewrite equations between exponential and logarithmic form

• analyze domain and range changes

• interpret inverse functions graphically and contextually

• use inverse functions to solve equations

Summary

Logarithmic functions:

• are defined as the inverses of exponential functions

• undo exponential growth and decay

• reverse domain and range

• provide powerful tools for equation solving and modeling