The Inverse of an Exponential Function

Idea

To undo an exponential you need whatever releases $x$ from a power. That is the logarithm, and it is the only thing that does the job.

Everything else in these questions is ordinary algebra.

Rule

$y = a^x \iff x = \log_a y$

$\log_a(a^x) = x$

Write $y$ for $f(x)$ , work towards $x$ , and swap the letters at the end.

How it is used

1. Undo the outside before the inside

An expression is built in layers, and an inverse peels them off in reverse order. The logarithm goes last, not first.

$f(x) = 2 \cdot 3^{x} + 5$	what to undo
the $+5$	subtract it
the $\times 2$	divide by it
the power	now apply $\log_3$

Reaching for the logarithm while a constant is still attached is the usual mistake. $\log_3(y - 5)$ is not the same as $\log_3\left(\dfrac{y - 5}{2}\right)$ .

2. The logarithm releases the whole exponent

It frees everything sitting in the power, not $x$ alone.

a^{x + k} = y \;\Rightarrow\; x + k = \log_a y

The $k$ then comes off by ordinary algebra, as a separate step.

3. Domain and range change places

An inverse reads the same pairs backwards, so what went in now comes out.

	domain	range
$f(x) = a^x$	every real $x$	$y > 0$
$f^{-1}(x) = \log_a x$	$x > 0$	every real $y$

The exponential's range being $y > 0$ is exactly why a logarithm's argument must be positive. It is one fact, seen from two sides.

4. The graphs are mirror images

Reflecting in $y = x$ swaps every coordinate pair, which is what an inverse does to the numbers.

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 2 · Blue: y = 2ˣ · Green: y = log_b(x) · Dashed: y = x. At x = 0.5, y = -1.

Worked example

Find the inverse of $f(x) = 2 \cdot 3^{x} + 5$ .

Remember

$\log_a(a^x) = x$ , so a logarithm releases a power undo the outside layers first, the power last an inverse is written in $x$ , so swap the letters at the end

Write $y$ for $f(x)$ .

y = 2 \cdot 3^{x} + 5

Subtract the constant. The power is still wrapped in a factor of $2$ .

y - 5 = 2 \cdot 3^{x}

Divide by that factor. Now the power stands alone.

\frac{y - 5}{2} = 3^{x}

Only now apply the logarithm, which releases the exponent.

x = \log_3\left(\frac{y - 5}{2}\right)

Swap the letters.

f^{-1}(x) = \log_3\left(\frac{x - 5}{2}\right)

Explore

The blue curve is $y = a^x$ and the green one is $y = \log_a x$ . Drag the probe and read both marked points: the same two numbers, in the opposite order.

That swap is what an inverse does. The dashed line is the mirror it happens across.