Logarithms and Inverse Functions

≈ 25 min

Undoing a function

The inverse $f^{-1}$ reverses what $f$ does: you swap $x$ and $y$ and make $y$ the subject.

Linear. For $f(x) = 2x - 6$ write $y = 2x - 6$ , swap: $x = 2y - 6$ , solve: $y = \dfrac{x+6}{2}$ . So $f^{-1}(x) = \dfrac{x+6}{2}$ .

Parabola. For $f(x) = 2x^2$ swapping gives $x = 2y^2$ , so $y = \pm\sqrt{\dfrac{x}{2}}$ — not a function. Restricting the domain to $x \ge 0$ gives the single branch $f^{-1}(x) = \sqrt{\dfrac{x}{2}}$ .

Exponential. For $f(x) = b^x$ (with $b > 0$ , $b \neq 1$ ) swapping gives $x = b^y$ , and the value of $y$ is defined to be the logarithm:

y = \log_b x \iff b^{\,y} = x

So the inverse of $y = 3^x$ is $y = \log_3 x$ .

Log laws

\log_b(MN) = \log_b M + \log_b N, \qquad \log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N, \qquad \log_b(M^k) = k\log_b M

and the change of base $\log_b x = \dfrac{\log x}{\log b}$ , which is what lets your calculator handle any base.

Worked example

A prepaid electricity meter's credit grows in a simulation as $3^x$ . Solve $3^x = 20$ :

x = \log_3 20 = \frac{\log 20}{\log 3} = \frac{1{,}30103}{0{,}47712} \approx 2{,}73

Check: $3^{2{,}73} \approx 20$ .

The number of households connected to a new solar micro-grid is modelled by $f(x) = 3^x$ , where $x$ is the number of years. Which function is $f^{-1}$ ?

$y = \log_3 x$

$y = 3^{-x}$

$y = x^3$

$y = \log_x 3$

A dam's algae population doubles every day and is modelled by $P = 2^x$ times its starting size. Solve for $x$ if $2^x = 50$ . Give your answer correct to two decimal places.

A taxi fare is $f(x) = 2x - 6$ rand for a trip of $x$ km beyond the base distance. Determine $f^{-1}(x)$ , writing your answer as an expression in $x$ .