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:
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undo exponential growth and decay
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solve exponential equations
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interpret models involving time, rate, and magnitude
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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 and are inverses if:
$$f(g(x))=x$$and$$g(f(x))=x$$
Graphically:
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Their graphs are reflections across the line y=x
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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:
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Domain: (−∞,∞)
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Range: (0,∞)
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Horizontal asymptote: y=0
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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
| Function | Domain | Range |
|---|---|---|
| $$y=a^x$$ | (−∞,∞) | (0,∞) |
| $$\log_{a}x$$ | (0,∞) | (−∞,∞) |
This reversal is a hallmark of inverse functions and is frequently tested conceptually on the AP exam.
Graphical Interpretation
Reflection Across y=x
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The graph of $$y=\log_{a}x$$ is the mirror image of $$y=a^x$$
Key Features of $$y=\log_{a}x$$
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Vertical asymptote: x=0
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x-intercept: (1,0)
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Increasing if a>1
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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
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Forgetting that logarithms require positive inputs
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Confusing $$\log_{a}x$$ with $$a^x$$
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Ignoring domain restrictions after inversion
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Treating inverse notation $$f^{-1}(x)$$ as a reciprocal
AP Exam Focus
Students should be able to:
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explain why logarithms are inverses of exponentials
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rewrite equations between exponential and logarithmic form
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analyze domain and range changes
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interpret inverse functions graphically and contextually
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use inverse functions to solve equations
Summary
Logarithmic functions:
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are defined as the inverses of exponential functions
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undo exponential growth and decay
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reverse domain and range
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provide powerful tools for equation solving and modeling