Definition: A logarithmic function is the inverse of an exponential function.
Example: $$f(x)=2^x \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, f^{-1}(x)=\log_2(x) \,\,\,\,\,$$
Logarithms: The Big Idea
The following two equations are equivalent: $$y=b^x \,\,\,\,\,\,\,\,\,\,\,\,\, x=\log_b(y)$$ Notice: both of the above equations say the same thing: "$y$ has $x$ factors of $b$ in it."
Also Notice: Logarithms allow us to solve equations where the exponent is the variable.
Example: $8=2^3$ is equivalent to the expression $3=\log_2(8)$.
Another Way of Putting It
The notation $$\log_b(x)$$ asks the question: "What power of $b$ is $x?$"
Example: What question does $\log_3(27)$ ask? What is the answer?
Example: What question does $\log_4(2)$ ask? What is the answer?
Examples: Evaluate the following logarithmic expressions.
$\log_2 16$
$\log_{4} 1$
$\log_{10} 10$
$\log_{3} \sqrt[3]{\frac{1}{27}}$
$\log_{10}0.0001$
Rewriting Equations
Remember: $y=b^x$ can always be rewritten as $x=\log_b(y)$, and vice versa.
Example: Convert the exponential equation to logarithmic form. $$4^{-3}=\frac{1}{64}.$$ Example: Convert the logarithmic equation to exponential form: $$\log_{3} \left(\frac{1}{9}\right)=-2.$$
Solving Equations
Remember: $y=b^x$ can always be rewritten as $x=\log_b(y)$, and vice versa.
Examples: Solve the following equations.
$\log_{4} x=2$
$\log_{11} x=-\frac{1}{2}$
$\log_{4}(2x-4)=3$
$\log_{x} 4=2$
Questions So Simple, They're Hard
Example: What question does $\log_3(3^3)$ ask? What is the answer?
Example: What question does $\log_{\frac{3}{4}}\left(\frac{3}{4}\right)^{-\frac{1}{3}}$ ask? What is the answer?
Consequence: A useful fact: $$\log_b (b^x)=x$$ Example: Evaluate $\log_{\gamma}\gamma^{\frac{1}{5}}$
Graphs of Logarithmic Functions
Complete the table of values $$\begin{array}{c|c}x & y \\ \hline1/4 & \\ 1 & \\ 4 & \\\end{array}$$ and use these points to sketch a graph of the function $f(x)=\log_4(x)$.