Key takeaway: A logarithm is the power you place on a chosen base to land on a target number, so writing log_b(x) = y says the same as b^y = x.
Grade 11 – Algebra 2F-LE.A.4Functions and Graphs
Definition of Logarithm
A logarithm is a question about powers: which exponent do I need? The expression \(\log_b x\) names the exponent that carries the base \(b\) up to \(x\). Because \(3^4=81\), the answer for \(\log_3 81\) is 4.
A logarithm is the inverse operation of an exponential, in the same way that subtraction undoes addition. That is why it rescues any equation whose unknown hides up in a power. The base must be positive and different from 1, and only positive numbers have a logarithm.
Formula and notation
The two statements below are equivalent, and you can rewrite one as the other at any moment:
\[\log_b x = y \iff b^{y}=x \quad (b>0,\ b\neq 1,\ x>0).\]
Here \(b\) is the base, \(x\) the argument and \(y\) the exponent. With no base shown, \(\log x\) usually means base 10.
Key facts
- The logarithm of 1 is 0 in every base, because b to the power 0 is 1.
- log_b(b) = 1, since any base to the first power is itself.
- Logarithms of numbers between 0 and 1 are negative.
- The graph of log_b(x) is the mirror image of b^x across the line y = x.
Logarithm example
Evaluate \(\log_2\frac{1}{32}\), then solve \(\log_7(x-1)=2\).
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Which power of 2 produces one thirty-second? A negative one, because \(2^{-5}\) equals that fraction. The value is \(-5\).
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Rewrite the equation in exponent form: \(7^2=x-1\).
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Then \(49=x-1\), so \(x=50\). Check: \(\log_7 49=2\).
\(\log_2\frac{1}{32}=\)\(-5\) and the equation gives \(x=50\).
Common mistake
Watch out
Darnell evaluates \(\log_5 125\) and answers 25, because \(5\times 25=125\).
Why it is wrong
The question asks for an exponent, not a factor. Find the power of 5 that equals 125: \(5^3=125\), so \(\log_5 125=3\).
Related terms and lessons
- Natural Logarithm (ln)
- Logarithm Rules
- Change of Base Formula
- Exponential Equation
- Euler's Number (e)
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