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AP Course

AP Precalculus

Study nine focused AP Precalculus units through topic lessons, original graph examples, quizzes, and practice sets.

Follow the College Board topic sequence across polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector, and matrix functions.

Lessons
Change in TandemRates of ChangeRates of Change in Linear and Quadratic FunctionsPolynomial Functions and Rates of ChangePolynomial Functions and Complex ZerosPolynomial Functions and End BehaviorRational Functions and End BehaviorRational Functions and ZerosRational Functions and Vertical AsymptotesRational Functions and HolesEquivalent Representations of Polynomial and Rational ExpressionsTransformations of FunctionsFunction Model Selection and Assumption ArticulationFunction Model Construction and ApplicationChange in Arithmetic and Geometric SequencesChange in Linear and Exponential FunctionsExponential FunctionsExponential Function ManipulationExponential Function Context and Data ModelingCompeting Function Model ValidationComposition of FunctionsInverse FunctionsLogarithmic ExpressionsInverses of Exponential FunctionsLogarithmic FunctionsLogarithmic Function ManipulationExponential and Logarithmic Equations and InequalitiesLogarithmic Function Context and Data ModelingSemi-log PlotsPeriodic PhenomenaSine, Cosine, and TangentSine and Cosine Function ValuesSine and Cosine Function GraphsSinusoidal FunctionsSinusoidal Function TransformationsSinusoidal Function Context and Data ModelingThe Tangent FunctionInverse Trigonometric FunctionsTrigonometric Equations and InequalitiesThe Secant, Cosecant, and Cotangent FunctionsEquivalent Representations of Trigonometric FunctionsTrigonometry and Polar CoordinatesPolar Function GraphsRates of Change in Polar FunctionsParametric FunctionsParametric Functions Modeling Planar MotionParametric Functions and Rates of ChangeParametrically Defined Circles and LinesImplicitly Defined FunctionsConic SectionsParametrization of Implicitly Defined FunctionsVectorsVector-Valued FunctionsMatricesThe Inverse and Determinant of a MatrixLinear Transformations and MatricesMatrices as FunctionsMatrices Modeling Contexts
Quizzes
Practice Problems Open a unit to choose from 58 topic-aligned practice sets.

Unit 4 · Topic 2.10

Inverses of Exponential Functions

Construct logarithmic inverses of exponential functions and connect their graphs through reflection. Develop the idea through symbolic, numerical, graphical, and contextual representations.

Learning Goals

  • Construct logarithmic inverses of exponential functions.
  • Connect reflected graphs, swapped points, and asymptotes.
  • Verify inverse identities.

1. Essential Structure

Construct logarithmic inverses of exponential functions and connect their graphs through reflection.

\[y=b^x\iff x=\log_b y\]

Read the formula together with its domain, units, starting input, and the interval length over which change is measured.

2. Core Ideas

  • The inverse of \(b^x\) is \(\log_bx\).
  • The exponential domain and range swap with the logarithmic range and domain.
  • The horizontal asymptote \(y=0\) reflects to vertical asymptote \(x=0\).
  • Multiplying a logarithm input by \(b\) adds 1 to its output.

3. Graph and Representation

Inverses of Exponential Functions example graphThe exponential and logarithmic graphs reflect across \(y=x\), swapping asymptotes and intercepts.
The exponential and logarithmic graphs reflect across \(y=x\), swapping asymptotes and intercepts.

Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.

4. Original Worked Example

Find the inverse of \(f(x)=2^x\).

The inverse is \(f^{-1}(x)=\log_2x\).

Check the result in the original representation and explain its meaning, including units when the quantities are contextual.

5. AP Reasoning Workflow

  1. Start with \(y=b^x\).
  2. Swap \(x\) and \(y\).
  3. Solve with logarithmic notation.
  4. Exchange domain and range.
  5. Verify both compositions.

A strong AP response shows the mathematical evidence first and then states a precise conclusion.

6. Extended Example and Application

Points \((-1,1/2)\), \((0,1)\), and \((3,8)\) on \(2^x\) become \((1/2,-1)\), \((1,0)\), and \((8,3)\) on \(\log_2x\).

7. Technology and Validation

Graph the inverse pair with \(y=x\) and use equal axis scales.

When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.

8. Why Every Basic Exponential Has an Inverse

For \(b>0\) and \(b\ne1\), the function \(f(x)=b^x\) is one-to-one: it is strictly increasing when \(b>1\) and strictly decreasing when \(0<b<1\). Therefore, each positive output comes from exactly one real input.

Feature\(f(x)=b^x\)Inverse function
Input setAll real numbersAll positive numbers
Output setAll positive numbersAll real numbers
Anchor pair\((0,1)\)\((1,0)\)
OperationRaise \(b\) to an exponentFind the required exponent

The inverse is named \(f^{-1}(x)=\log_bx\). This notation represents a logarithmic function, not the reciprocal \(1/b^x=b^{-x}\).

9. Derive the Logarithmic Inverse Algebraically

Begin with the exponential function and reverse its input and output roles:

\[y=b^x.\]
  1. Exchange \(x\) and \(y\): \(x=b^y\).
  2. Ask which exponent on \(b\) produces \(x\).
  3. Write that exponent with logarithmic notation: \(y=\log_bx\).
\[f(x)=b^x\quad\Longleftrightarrow\quad f^{-1}(x)=\log_bx.\]

For example, the inverse of \(f(x)=5^x\) is \(f^{-1}(x)=\log_5x\). The logarithm requires \(x>0\) because the exponential function never outputs zero or a negative number.

10. Construct the Inverse Table by Swapping Coordinates

Choose exponent inputs that produce recognizable powers, then exchange every ordered pair.

\(x\)\(3^x\)Reflected inverse pairLogarithmic statement
-2\(1/9\)\((1/9,-2)\)\(\log_3(1/9)=-2\)
-1\(1/3\)\((1/3,-1)\)\(\log_3(1/3)=-1\)
01\((1,0)\)\(\log_3(1)=0\)
13\((3,1)\)\(\log_3(3)=1\)
29\((9,2)\)\(\log_3(9)=2\)

The inverse table's inputs are all positive. Negative logarithmic outputs are valid; they correspond to exponential outputs between 0 and 1.

11. Reflect Domains, Intercepts, and Asymptotes

The graphs of \(y=b^x\) and \(y=\log_bx\) are reflections across the identity line \(y=x\). Reflection exchanges horizontal and vertical information.

Exponential featureReflected logarithmic feature
Domain \(( -\infty,\infty)\)Range \(( -\infty,\infty)\)
Range \((0,\infty)\)Domain \((0,\infty)\)
y-intercept \((0,1)\)x-intercept \((1,0)\)
Horizontal asymptote \(y=0\)Vertical asymptote \(x=0\)
No x-interceptNo y-intercept

The logarithmic graph approaches \(x=0\) only from the right because its domain contains positive inputs. It never crosses the vertical asymptote.

Graphing check: use equal axis scales; otherwise a true reflection across \(y=x\) may appear distorted.

12. Compare Growth and Decay Bases

Inversion preserves monotonic direction. An increasing one-to-one function has an increasing inverse, and a decreasing one-to-one function has a decreasing inverse.

BaseExponential behaviorLogarithmic behaviorSample inverse pair
\(b=4\)\(4^x\) increases\(\log_4x\) increases\(4^2=16\leftrightarrow\log_4(16)=2\)
\(b=1/4\)\((1/4)^x\) decreases\(\log_{1/4}x\) decreases\((1/4)^{-2}=16\leftrightarrow\log_{1/4}(16)=-2\)

For \(b>1\), larger positive inputs produce larger logarithmic outputs. For \(0<b<1\), larger positive inputs produce smaller logarithmic outputs. Both inverse graphs still pass through \((1,0)\).

13. Connect Scaled Logarithms to Exponential Inverses

The general-form logarithmic function \(g(x)=a\log_bx\), where \(a\ne0\), is also an inverse of a related exponential function. Start with

\[y=a\log_bx.\]

Divide by \(a\) and rewrite in exponential form:

\[\frac{y}{a}=\log_bx\quad\Longrightarrow\quad x=b^{y/a}.\]

Therefore,

\[g^{-1}(x)=b^{x/a}.\]

Equivalently, if \(f(x)=b^{cx}\) with \(c\ne0\), then

\[f^{-1}(x)=\frac1c\log_bx.\]

Example: the inverse of \(f(x)=2^{3x}\) is \(f^{-1}(x)=\frac13\log_2x\). The factor \(1/3\) reverses the input scaling by 3 in the exponential function.

14. Verify the Inverse and Interpret Multiplicative Change

Inverse functions undo one another on their proper domains:

\[\log_b(b^x)=x\quad(x\in\mathbb R),\qquad b^{\log_bx}=x\quad(x>0).\]

For \(f(x)=2^{3x}\) and \(g(x)=\frac13\log_2x\),

\[g(f(x))=\frac13\log_2(2^{3x})=x,\]
\[f(g(x))=2^{3(\frac13\log_2x)}=2^{\log_2x}=x,\qquad x>0.\]

The inverse relationship also explains logarithmic change. If \(L(x)=\log_bx\), then multiplying the input by \(b\) adds one to the output:

\[L(bx)=L(x)+1.\]

This statement can be understood without using logarithm properties: if \(L(x)=k\), then \(x=b^k\), so \(bx=b^{k+1}\) and \(L(bx)=k+1\).

AP-ready explanation: identify the swapped coordinates, exchange domain and range, reflect the asymptote, and use composition or the exponential-logarithmic definition to verify the inverse.

15. Common Errors

  • Do not forget that the logarithmic inverse has domain \(x>0\).
  • Keeping the exponential asymptote horizontal after inversion.
  • Using logarithm inputs outside \(x>0\).
  • Giving a numerical result without a domain check, units, or interpretation.

Key Takeaways

  • Construct logarithmic inverses of exponential functions and connect their graphs through reflection.
  • Core relationship: \(y=b^x\iff x=\log_b y\)
  • Error check: Do not forget that the logarithmic inverse has domain \(x>0\).
Checkpoint · Topic 2.10
  1. Swap three points between \(3^x\) and its inverse.
  2. Compare their domains and ranges.
  3. Explain the output effect of multiplying a logarithm input by its base.