Key takeaway: The logarithm rules turn products into sums, quotients into differences and powers into multiples, so log_b(xy) = log_b x + log_b y and log_b(x^n) = n·log_b x.
Grade 11 – Algebra 2F-LE.A.4Functions and Graphs
Definition of Logarithm Rules
The logarithm rules are three shortcuts that mirror the laws of exponents. Because a logarithm is an exponent, multiplying numbers adds their logs, dividing numbers subtracts their logs, and raising a number to a power multiplies its log by that power.
These logarithm rules let you break one complicated expression into simple pieces (expanding) or fuse many pieces into one (condensing). They also reduce hard equations to linear ones. All three require the same base throughout, and every argument must stay positive.
Formula and notation
For a base \(b>0\), \(b\neq 1\), and positive numbers \(x\) and \(y\):
\[\log_b(xy)=\log_b x+\log_b y,\quad \log_b\frac{x}{y}=\log_b x-\log_b y,\quad \log_b(x^n)=n\log_b x.\]
Two special values go with them: \(\log_b 1=0\) and \(\log_b b=1\).
Key facts
- The product rule adds, the quotient rule subtracts, and the power rule multiplies.
- Each rule is the logarithmic twin of an exponent law, such as b^m · b^n = b^(m+n).
- There is no rule for the log of a sum or a difference.
- Expanding and condensing are the same rules read left to right or right to left.
Logarithm Rules example
Write \(2\log_6 3+\log_6 8-\log_6 2\) as a single logarithm and evaluate it.
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Power rule on the first term: \(2\log_6 3=\log_6 9\).
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Product rule for the sum: \(\log_6 9+\log_6 8=\log_6 72\).
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Quotient rule for the difference: \(\log_6 72-\log_6 2=\log_6 36\).
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Since \(6^2=36\), the value is 2.
The expression equals \(\log_6 36=\)2.
Common mistake
Watch out
Ingrid simplifies \(\frac{\log_3 20}{\log_3 4}\) as \(\log_3(20-4)=\log_3 16\).
Why it is wrong
The quotient rule applies to the log of a quotient, not to a quotient of two logs. The fraction \(\frac{\log_3 20}{\log_3 4}\) is not equal to \(\log_3 16\): the first is about 2.16, the second about 2.52.
Related terms and lessons
- Algebra 2 · Logarithms and Logarithmic FunctionsLessonWorksheetQuiz
- Algebra 2 · Solving Exponential and Logarithmic EquationsLessonWorksheetQuiz
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