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Logarithmic Functions Solutions

A logarithmic function is the inverse of an exponential function for base b (when $b > 0, b \neq 1$ ). 
To find the inverse of an exponential function:

Switch x and y in the equation. 
Take $\begin{array}{rcl} \log_{b} \end{array}$ of both sides. 
Solve for y.

$y = {(b)}^{x} x = {(b)}^{y}$
  $\begin{array}{rcl} \log_{b} x & = & \log_{b} {(b)}^{y} \\ \log_{b} x & = & y \\ y & = & \log_{b} x, x > 0 \end{array}$

Example 1

Write the inverse of the given function.

$f (x) = {(\frac{1}{3})}^{x}$

$b = \frac{1}{3} f^{- 1} (x) = \log_{\frac{1}{3}} x$

$r (x) = e^{x}$

$b = e r^{- 1} (x) = \log_{e} x r^{- 1} (x) = \ln x$

$h (x) = {(\frac{1}{3})}^{- x}$

${(\frac{1}{3})}^{- x} = 3^{x} b = 3 h^{- 1} (x) = \log_{3} x$

Example 2

Write the inverse of the given function.

$g (x) = 6^{x} - 4$

Implement

$\begin{matrix} y = 6^{x} - 4 \\ x = 6^{y} - 4 \\ x + 4 = 6^{y} \\ \log_{6} (x + 4) = \log_{6} {(6)}^{y} \\ \log_{6} (x + 4) = y \\ y = \log_{6} (x + 4) \end{matrix}$

Explain

Write as y =
Switch x and y
Take $\begin{array}{rcl} \log_{b} \end{array}$ of both sides
Solve for y