Then, expressions like $f(3)=2^3$ or $f(9)=2^9$ are asking “What do I get when I multiply $2$ by itself three times or nine times?” or rather “If I raise 2 to the 3rd power, or to the 9th power, what do I get?”

Logarithms ask the opposite question. They start with an equation like

$$2^x=64$$

and ask the question “what exponent $x$ do I need to raise $2$ to so that $2^x=64$? In other words, we are not looking for the result of applying the power to some base like up at the top of this page, but rather we are looking for the exponent that yields a given output result.

Logarithms

Let $a,x$ be positive real numbers so that $a\neq 1$, and let $y$ be any real number. Then,
$$a^x=y\ \text{is equivalent to}\ \log_{a}{y}=x$$
where the expression $\log_{a}{y}$ represents the exponent ($x$) that we need to raise $a$ to in order to produce an output of $y$.

In short, logarithms are a notation (and a function type) that we use with exponential expressions when provided an output but are not given the exponent. Logarithms find (or rather, represent) the missing exponent.

Lesson Content

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Evaluating Logarithmic Functions at Given x-Values

Graphing and Transforming Logarithmic Functions

Log Laws and Change-of-Base Formula

Solving Logarithmic and Exponential Equations

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