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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 2.12

Logarithmic Function Manipulation

Expand and condense logarithmic expressions using product, quotient, and power properties. Develop the idea through symbolic, numerical, graphical, and contextual representations.

Learning Goals

  • Expand and condense logarithms.
  • Use product, quotient, power, and change-of-base properties.
  • Preserve domains and absolute values.

1. Essential Structure

Expand and condense logarithmic expressions using product, quotient, and power properties.

\[\log_b(MN)=\log_bM+\log_bN\]

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

2. Core Ideas

  • Products become sums and quotients become differences.
  • Powers become coefficients on a valid shared domain.
  • \(\log_bx=\ln x/\ln b\) shows that bases differ by vertical scale.
  • There is no logarithm property for splitting a sum.

3. Graph and Representation

Logarithmic Function Manipulation example graphEquivalent logarithmic expressions produce the same outputs on their shared domain.
Equivalent logarithmic expressions produce the same outputs on their shared domain.

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

Rewrite \(\ln(x^2)\) for real \(x\ne0\).

\(\ln(x^2)=2\ln|x|\).

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

5. AP Reasoning Workflow

  1. Factor expressions when useful.
  2. Record positivity restrictions.
  3. Apply one property at a time.
  4. Use absolute values where required.
  5. Compare the original and rewritten domains.

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

6. Extended Example and Application

For \(x>3\), \(\ln((x^2-9)/\sqrt{x})=\ln(x-3)+\ln(x+3)-\tfrac12\ln x\).

7. Technology and Validation

Use change of base for approximations and keep exact forms until rounding is necessary.

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

8. Connect Logarithm Laws to Exponent Laws

Logarithm laws follow from the laws of exponents because logarithms record exponents. For a valid base \(b\) and positive \(M,N\):

Exponent lawCorresponding logarithm lawOperation change
\(b^m b^n=b^{m+n}\)\(\log_b(MN)=\log_bM+\log_bN\)Product becomes sum
\(b^m/b^n=b^{m-n}\)\(\log_b(M/N)=\log_bM-\log_bN\)Quotient becomes difference
\((b^m)^r=b^{mr}\)\(\log_b(M^r)=r\log_bM\)Power becomes coefficient

If \(M=b^m\) and \(N=b^n\), then \(MN=b^{m+n}\), so the exponent required to produce \(MN\) is \(m+n\). This reasoning explains the product law.

No sum law: exponents provide no rule turning \(b^m+b^n\) into \(b^{m+n}\). Therefore, \(\log_b(M+N)\) cannot be split into two logarithms.

9. Expand a Logarithmic Expression

Expansion separates products, quotients, and powers while preserving the original structure. For \(x>3\) and \(y>0\), expand

\[\log_5\left(\frac{25x^3\sqrt{y}}{(x-3)^2}\right).\]
  1. Use the quotient law.
  2. Separate numerator products.
  3. Move exponents and roots in front as coefficients.
  4. Evaluate \(\log_5(25)=2\).
\[2+3\log_5x+\frac12\log_5y-2\log_5(x-3).\]

The stated restrictions make every displayed logarithm valid. Factor before expanding when useful, but cancel common factors first and retain any values excluded by the original expression.

10. Condense Several Terms into One Logarithm

Condensing reverses the logarithm laws. Apply coefficients as powers first, combine sums as products, and combine differences as quotients.

For \(x>0\),

\[3\ln x+\frac12\ln(x+2)-\ln7\]

becomes

\[\ln(x^3)+\ln\big((x+2)^{1/2}\big)-\ln7\]
\[=\ln\left(\frac{x^3\sqrt{x+2}}{7}\right).\]

Only logarithms with the same base can be combined directly. For example, \(\log_2x+\log_3y\) is not a product-law pair until one expression is rewritten to a common base.

11. Preserve Domains and Use Absolute Values

Algebraically similar logarithmic forms may have different real domains. The equation

\[\ln(x^2)=2\ln|x|,\qquad x\ne0,\]

is valid for both positive and negative \(x\). Writing \(2\ln x\) would incorrectly discard all negative inputs.

Consider

\[\ln\big((x-1)(x+2)\big).\]

The original argument is positive when \(x<-2\) or \(x>1\). On that original domain,

\[\ln|x-1|+\ln|x+2|\]

preserves the values on both intervals, whereas \(\ln(x-1)+\ln(x+2)\) retains only \(x>1\). The absolute-value expression by itself is also defined on \((-2,1)\), where the original logarithm is not, so the original restriction must remain attached to the rewritten form.

Before rewritingRequired check
Logarithm of a productThe whole product must be positive
Sum of separate logarithmsEvery separate argument must be positive
Even power inside a logarithmThe base expression may be positive or negative, but not zero
Cancellation inside an argumentRetain values excluded by the original denominator

12. Interpret Logarithm Laws as Graph Transformations

The product and power laws show that some transformations of a logarithmic graph have equivalent descriptions.

Input scaling and output translation: for \(k>0\),

\[\log_b(kx)=\log_bx+\log_bk.\]

Multiplying the input by \(k\) produces a horizontal dilation by factor \(1/k\), or equivalently a vertical translation by \(\log_bk\). For example,

\[\log_2(8x)=\log_2x+3.\]

Input power and output dilation: on a shared valid domain,

\[\log_b(x^r)=r\log_bx.\]

Raising the input to the power \(r\) corresponds to scaling every logarithmic output by \(r\). These are equivalent descriptions of the same function, not transformations applied one after another.

13. Change the Base and Recognize the Natural Logarithm

To rewrite \(\log_bM\) using any valid new base \(n\), use

\[\log_bM=\frac{\log_nM}{\log_nb}.\]

Choosing base \(e\) gives

\[\log_bM=\frac{\ln M}{\ln b},\]

where \(\ln M=\log_eM\) and \(e\approx2.71828\). Choosing base 10 gives \(\log_bM=\log M/\log b\).

Example:

\[\log_7(40)=\frac{\ln40}{\ln7}\approx1.8957.\]

The estimate is reasonable because \(7^1<40<7^2\). Keep the quotient exact during algebra and round only the final numerical result.

For a fixed argument, changing the logarithm's base multiplies all outputs by a constant. Thus all parent logarithmic graphs are vertical dilations or reflections of one another.

14. Complete AP-Style Manipulation Example

Suppose \(p=\log_2(3)\) and \(q=\log_2(5)\). Express

\[\log_2\left(\frac{45}{8}\right)\]

in terms of \(p\) and \(q\).

  1. Factor: \(45=3^2\cdot5\) and \(8=2^3\).
  2. Use the quotient law: \(\log_2(45)-\log_2(8)\).
  3. Use product and power laws: \(2\log_2(3)+\log_2(5)-3\log_2(2)\).
  4. Substitute \(p\), \(q\), and \(\log_2(2)=1\).
\[\log_2\left(\frac{45}{8}\right)=2p+q-3.\]

Reverse check:

\[2p+q-3=\log_2(3^2)+\log_2(5)-\log_2(2^3)=\log_2\left(\frac{45}{8}\right).\]
AP-ready reasoning: name the property, keep the base consistent, preserve the domain, and distinguish a valid product inside a logarithm from an unsplittable sum.

15. Common Errors

  • Do not rewrite \(\log(M+N)\) as \(\log M+\log N\).
  • Splitting \(\log(M+N)\).
  • Writing \(\ln(x^2)=2\ln x\) for negative \(x\) instead of \(2\ln|x|\).
  • Giving a numerical result without a domain check, units, or interpretation.

Key Takeaways

  • Expand and condense logarithmic expressions using product, quotient, and power properties.
  • Core relationship: \(\log_b(MN)=\log_bM+\log_bN\)
  • Error check: Do not rewrite \(\log(M+N)\) as \(\log M+\log N\).
Checkpoint · Topic 2.12
  1. Expand \(\log_2((x-1)^3/(x+4))\) for \(x>1\).
  2. Condense \(2\ln x-\tfrac12\ln(x+1)\).
  3. Explain the graphical meaning of change of base.