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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 2.13

Exponential and Logarithmic Equations and Inequalities

Solve equations and inequalities using common bases, inverses, monotonicity, and domain checks. Develop the idea through symbolic, numerical, graphical, and contextual representations.

Learning Goals

  • Solve exponential and logarithmic equations and inequalities.
  • Construct inverses of transformed functions.
  • Reject extraneous or noncontextual solutions.

1. Essential Structure

Solve equations and inequalities using common bases, inverses, monotonicity, and domain checks.

\[b^{u}=b^{v}\Rightarrow u=v\quad(b>0,b\ne1)\]

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

2. Core Ideas

  • Use common bases when possible; otherwise isolate and take logarithms.
  • Record logarithm argument restrictions before combining terms.
  • Increasing functions preserve inequality direction; decreasing functions reverse it.
  • Inverse construction reverses transformations in reverse order.

3. Graph and Representation

Exponential and Logarithmic Equations and Inequalities example graphAn intersection graph confirms where the two sides have equal output values.
An intersection graph confirms where the two sides have equal output values.

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

Solve \(3^{2x}=27\).

Since \(27=3^3\), \(2x=3\) and \(x=\frac{3}{2}\).

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

5. AP Reasoning Workflow

  1. State restrictions.
  2. Isolate or combine the nonlinear expression.
  3. Apply an inverse operation.
  4. Solve.
  5. Check in the original statement and interpret.

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

6. Extended Example and Application

For \(\log_2(x-1)+\log_2(x-3)=3\), the domain is \(x>3\). Solving \((x-1)(x-3)=8\) gives 5 and \(-1\), but only \(x=5\) is valid.

7. Technology and Validation

Use graph intersections when symbolic solutions are inaccessible, then verify domain and report appropriate precision.

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

8. Choose an Efficient Solving Strategy

Identify the equation's structure before manipulating it.

StructurePreferred first moveWhy it helps
Exponential terms can share one baseRewrite with a common baseOne-to-one behavior lets you equate exponents
One isolated exponential remainsApply a logarithmThe logarithm brings the exponent down
Powers such as \(b^{2x}\) and \(b^x\)Substitute \(u=b^x>0\)The equation becomes polynomial
Several same-base logarithmsCombine themOne logarithm can be rewritten exponentially
Two same-base logarithms are equalEquate their argumentsLogarithmic functions are one-to-one
No accessible symbolic solutionUse graph or numerical intersectionTechnology can estimate the common output point

Before solving a logarithmic equation, record all argument restrictions. After solving, substitute each candidate into the original equation and apply any contextual limits.

9. Solve Exponential Equations with a Common Base

For \(b>0\) and \(b\ne1\),

\[b^U=b^V\quad\Longleftrightarrow\quad U=V.\]

Solve

\[9^{x-1}=27^{2x+1}.\]

Rewrite both sides with base 3:

\[(3^2)^{x-1}=(3^3)^{2x+1}\]
\[3^{2x-2}=3^{6x+3}.\]

Thus \(2x-2=6x+3\), so

\[x=-\frac54.\]

Do not set unlike bases' exponents equal. The one-to-one step applies only after both sides represent powers of the same valid base.

10. Use Logarithms When Bases Do Not Match

When no convenient common base exists, isolate the positive exponential expression and take a logarithm of both sides. For

\[7(3^{2x})=50,\]

divide by 7 and apply the natural logarithm:

\[3^{2x}=\frac{50}{7},\qquad 2x\ln3=\ln\left(\frac{50}{7}\right).\]
\[x=\frac{\ln(50/7)}{2\ln3}\approx0.895.\]

The exact logarithmic form should be retained until the final requested approximation.

Logarithms also reveal useful equivalent exponential forms. For positive \(b,c\),

\[\left(b^x\right)^{\log_b c}=b^{x\log_b c}=c^x.\]

This identity can replace an awkward exponential base by a more useful one without changing the function.

11. Solve Logarithmic Equations and Reject Extraneous Roots

Solve

\[\log_2(x+1)+\log_2(x-3)=3.\]

First record the domain: \(x+1>0\) and \(x-3>0\), so \(x>3\). Combine and convert to exponential form:

\[\log_2((x+1)(x-3))=3\Longrightarrow(x+1)(x-3)=8.\]
\[x^2-2x-11=0\Longrightarrow x=1\pm2\sqrt3.\]

Only \(x=1+2\sqrt3\) lies in the original domain. The other algebraic root makes \(x-3\) negative and is extraneous.

One-to-one form: if \(\log_bM=\log_bN\), then \(M=N\), but both \(M\) and \(N\) must remain positive.

12. Solve Exponential and Logarithmic Inequalities

Increasing functions preserve order; decreasing functions reverse it.

FunctionDirectionComparison rule
\(b^x\), \(b>1\)IncreasingExponent inequality keeps its direction
\(b^x\), \(0<b<1\)DecreasingExponent inequality reverses
\(\log_bx\), \(b>1\)IncreasingArgument inequality keeps its direction
\(\log_bx\), \(0<b<1\)DecreasingArgument inequality reverses

Exponential example:

\[\left(\frac12\right)^{x+1}>\left(\frac12\right)^3\Longrightarrow x+1<3\Longrightarrow x<2.\]

Logarithmic example: solve \(\log_{1/3}(2x-1)\ge-2\). The domain requires \(x>1/2\). Since the base is less than 1,

\[2x-1\le\left(\frac13\right)^{-2}=9\Longrightarrow x\le5.\]

The solution is \((1/2,5]\).

13. Use Substitution for Exponential Quadratic Form

Expressions involving \(b^{2x}\) and \(b^x\) often behave like a quadratic. Solve

\[4^x-5(2^x)+4=0.\]

Because \(4^x=(2^x)^2\), let \(u=2^x\). Exponential outputs are positive, so \(u>0\).

\[u^2-5u+4=0\Longrightarrow(u-1)(u-4)=0.\]

Thus \(u=1\) or \(u=4\). Returning to the original variable:

\[2^x=1\Longrightarrow x=0,\qquad2^x=4\Longrightarrow x=2.\]

If factoring produced a negative value of \(u\), that branch would be rejected because \(b^x\) is always positive. This positivity check is separate from ordinary polynomial solving.

14. Construct Inverses of Transformed Exponential and Logarithmic Functions

Reverse transformations in reverse order. For

\[f(x)=a b^{x-h}+k,\qquad a\ne0,\]

subtract \(k\), divide by \(a\), apply \(\log_b\), and add \(h\):

\[f^{-1}(x)=h+\log_b\left(\frac{x-k}{a}\right),\qquad\frac{x-k}{a}>0.\]

For example, if \(f(x)=4(2^{x-3})-5\), then

\[f^{-1}(x)=3+\log_2\left(\frac{x+5}{4}\right),\qquad x>-5.\]

Likewise, for

\[g(x)=a\log_b(x-h)+k,\]

the inverse is

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

The original vertical shift becomes part of the inverse exponent, while the original horizontal shift becomes an outside addition. Verify by composition and exchange the original domain and range.

15. Common Errors

  • Check every logarithmic solution in the original domain.
  • Failing to reverse an inequality for a decreasing function.
  • Keeping roots that make a logarithm argument nonpositive.
  • Giving a numerical result without a domain check, units, or interpretation.

Key Takeaways

  • Solve equations and inequalities using common bases, inverses, monotonicity, and domain checks.
  • Core relationship: \(b^{u}=b^{v}\Rightarrow u=v\quad(b>0,b\ne1)\)
  • Error check: Check every logarithmic solution in the original domain.
Checkpoint · Topic 2.13
  1. Solve \(5^{2x-1}=17\) exactly and approximately.
  2. Solve \(\ln x-\ln(x-2)=\ln3\).
  3. Find the inverse of \(4(2^{x-3})-5\) and state its domain.