Study guide
9.3 Logarithms and Logarithmic Properties
ACCUPLACER Advanced Algebra and Functions Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
9.3 Logarithms and Logarithmic Properties
Key Takeaways Explain This Topic AI
Definition: log_b a = c if and only if b^c = a, with a > 0, b > 0, and b ≠ 1. So log_2 8 = 3 because 2^3 = 8.
Skills Insight 276–300 evaluates logarithmic expressions: log_3 81 = 4, log_5(1/25) = −2, log_10 1000 = 3, and ln(e^4) = 4.
Product, quotient, and power rules: log(xy) = log x + log y, log(x/y) = log x − log y, and log(x^n) = n log x. Expand log(xy^2) = log x + 2 log y.
Common log is log_10 (written log) and natural log is log_e (written ln), at recognition level on AAF.
Change of base is an optional exact-form tool: log_b a = log a / log b = ln a / ln b. Logs and exponentials are inverse functions.
Last updated: August 2026
9.3 Logarithms and Logarithmic Properties
A logarithm is an exponent. Table 11 pairs logarithmic equations with exponential ones because the two languages name the same relationship. This section is the translation: log_b a = c if and only if b^c = a , plus the product, quotient, and power rules you need to expand or condense, and the domain restrictions AAF will enforce when you later solve. Skills Insight band 276–300 specifically includes evaluating logarithmic expressions and equations .
Graphing the exponential parent was Graphs of Exponential Functions . Using these properties to solve equations is Solving Exponential and Logarithmic Equations . Inverse-function language in general sits in Function Operations, Composition, and Inverses . Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra .
- Definition: log_b a = c iff b^c = a
- The statement
- log_b a = c ⇔ b^c = a
means: the logarithm log_b a is the exponent you put on base b to get a .
b is the base of the log (and of the matching exponential).
a is the argument (the number you are “taking the log of”).
c is the value of the log — an exponent.
Worked evaluations (Skills Insight 276–300)
log_2 8 = 3 , because 2^3 = 8 . That is the worked identity this section must make automatic. Ask “2 to what power is 8?” not “2 times what is 8?”
- Logarithmic form Exponential form Value
- log_2 8 2^? = 8 3
- log_3 81 3^? = 81 4 ( 3^4 = 81 )
- log_5 (1/25) 5^? = 1/25 −2 ( 5^{−2} = 1/25 )
- log_7 1 7^? = 1 0 ( b^0 = 1 )
- log_4 4 4^? = 4 1 ( b^1 = b )
- log_10 1000 10^? = 1000 3
- log_2 (1/8) 2^? = 1/8 −3
Negative values of the log are normal: they mean a negative exponent , which is a reciprocal, not a negative argument. The argument 1/25 is still positive.
Identities that fall out of the definition:
- log_b 1 = 0
- log_b b = 1
- log_b (b^k) = k
- b^{log_b a} = a for a > 0
The last two say that exponential and logarithm with the same base are inverse functions . Applying one undoes the other, the same way f and f^{−1} undo each other.
Common log and natural log, at recognition level
AAF expects you to recognize two named logs, not to compute messy decimals by hand.
- Name Notation Base Recognition facts
- Common log log x (no base written) 10 log 1000 = 3 , log 0.01 = −2
- Natural log ln x e ln(e^4) = 4 , ln 1 = 0 , e^{ln 7} = 7
On AAF, log without a base is base 10, not base 2. If a stem writes log_2 , the 2 is required. ln is always base e . You do not need a numeric value of e ≈ 2.718 to evaluate ln(e^4) ; the inverse identity does the work.
A calculator icon may appear on some items (College Board calculator policy). Handheld calculators are not allowed except with an approved accommodation. Prefer exact rewrite: log_2 32 = 5 is better than a decimal approximation of ln 32 / ln 2 .
Product, quotient, and power rules
Let every log below have the same base b > 0 , b ≠ 1 , and let arguments be positive.
- Rule Formula In words
- Product log_b (xy) = log_b x + log_b y Log of a product is a sum of logs
- Quotient log_b (x/y) = log_b x − log_b y Log of a quotient is a difference
- Power log_b (x^n) = n log_b x An exponent inside comes out front
These are the exponential laws rewritten. Because b^c · b^d = b^{c+d} , adding exponents (logs) multiplies the arguments.
Worked: expand log(xy^2)
Expand log(xy^2) (any legal base, including common log).
Product rule: log(xy^2) = log x + log(y^2) .
Power rule on the second term: log(y^2) = 2 log y .
Result: log x + 2 log y .
That is the expansion this section must produce without hesitation. Traps:
log x + log y^2 is not fully expanded (the power rule still applies).
2 log(xy) is log((xy)^2) = log(x^2 y^2) , a different expression.
log x · 2 log y multiplies logs; the product rule adds them.
log(x + y^2) cannot be split at all. There is no sum rule of the form log(x + y) = log x + log y .
Condense the other direction: 2 log_5 m − log_5 n = log_5 (m^2 / n) . Bring coefficients back as exponents, then combine a difference as a quotient.
Another expansion: ln(8 / w^3) = ln 8 − ln(w^3) = ln 8 − 3 ln w . If you want, ln 8 = ln(2^3) = 3 ln 2 , so 3 ln 2 − 3 ln w = 3 ln(2/w) .
Change of base, as an optional exact-form tool
If AAF leaves an answer as a log with an inconvenient base, change of base rewrites it in common or natural logs:
log_b a = (log a) / (log b) = (ln a) / (ln b)
Example: log_3 11 = ln 11 / ln 3 = log 11 / log 3 . That is an exact form, not a rounded decimal. You will use it in the next section when 3^{4x} = 11 produces x = (log_3 11)/4 , which is also x = ln 11 / (4 ln 3) .
Change of base is optional when the original log already matches the options. Do not change base just to introduce a calculator. Do not write log_3 11 = log 3 / log 11 — that is the reciprocal, log_11 3 .
Domain: a > 0, b > 0, b ≠ 1
The real logarithm log_b a requires:
argument a > 0 (you cannot take a real log of 0 or of a negative);
base b > 0 and b ≠ 1 (the same restrictions as an exponential base).
So log_2 (x − 3) needs x − 3 > 0 , hence x > 3 . That restriction is not optional decoration; it is why solving log equations in the next section includes a domain check . log(0) and log(−5) are not real. log_1 5 is not defined. log_{−2} 8 is not an AAF real-valued log.
The range of y = log_b x is all real numbers: logs can be negative, zero, or positive, as log_2 (1/8) = −3 already showed. Graphically, y = log_b x is the reflection of y = b^x across the line y = x . The exponential’s horizontal asymptote y = 0 becomes the log’s vertical asymptote x = 0 . Domain and range swap, which is what inverses do.
Inverse pair in function notation
If f(x) = b^x , then f^{−1}(x) = log_b x . So f(3) = 8 would mean f^{−1}(8) = 3 when b^3 = 8 . The functions chapter’s practice with inverse-function evaluation reappears here, and this time the inverse has a name: logarithm.
A compact checklist before any expansion or evaluation:
Convert to exponential if you are evaluating a single log.
Apply product / quotient / power only to products, quotients, and powers , never to sums.
Keep a > 0 .
Common log is base 10; ln is base e .
Once those four are automatic, solving — same-base exponents, converting 3^{4x} = 11 into a log, and exponentiating a log equation — is the last section of this chapter.