Mathematics 9709 · Pure Mathematics 3

e^x and ln x: inverse twins, graphs, and the e/ln cancel

$e\approx2.718$ is a special base (its digits never stop or repeat).
Its exponential $y=e^x$ and its logarithm $y=\ln x$ are inverse functions: each one undoes the other.
That cancel is behind every solving and straightening move in this chapter.

$e^{\ln x}=x$ and $\ln(e^x)=x$ : wrap one around the other and they cancel. That cancel is the “undo” button you press to free $x$ . (textbook §2.7, Key point 2.5)

Two curves, mirrored in y = x

Because they are inverses, the graph of $y=\ln x$ is just $y=e^x$ flipped across the line $y=x$ .
Learn the two shapes once and you won't sketch them wrong.

y = e^x and y = ln x are mirror images in y = x. e^x goes through (0, 1); ln x goes through (1, 0).

Curve	Passes through	Asymptote	Lives where
$y=e^x$	$(0,1)$	$y=0$ (from above)	always $>0$
$y=\ln x$	$(1,0)$	$x=0$ (vertical)	only $x>0$

Flipping in $y=x$ swaps the axes, so every feature swaps too. The point $(0,1)$ becomes $(1,0)$ . The flat asymptote $y=0$ becomes the upright asymptote $x=0$ . So $\ln x$ drops down to $-\infty$ as $x$ gets close to $0$ from above.

The cancel, used for real

These two identities are how you free $x$ .

Worked example

Quick cancels

$\ln(e^x)=15\Rightarrow x=15$ : the $\ln$ and $e$ cancel.
$e^{3\ln x}=64$ : pull the $3$ up first, $e^{\ln x^3}=x^3$ , so $x^3=64\Rightarrow x=4$ .

(textbook Exercise 2G)

Common mistake

They only cancel when one is wrapped straight around the other.

e^{3\ln x}

is not

3x

. The

3

is a power on the inside, so it acts as a power first (

x^3

), then cancels.

Finding an inverse, the e/ln way

P3 likes “find $f^{-1}$ ” for things like $f(x)=5e^x+2$ . Set $y=5e^x+2$ , get $e^x$ on its own ( $e^x=\frac{y-2}{5}$ ), then take $\ln$ to free the $x$ : $x=\ln\!\frac{y-2}{5}$ . Swap the letters for the final answer $f^{-1}(x)=\ln\!\frac{x-2}{5}$ . The $\ln$ just undoes the $e$ . (textbook Exercise 2G Q12)

Where the shapes pay off

Knowing these two curves lets you answer the “sketch a pair of graphs to show there is only one root” questions. Draw the $y=e^x$ and $y=\ln x$ shapes and count where they cross.
Actually finding the root is the numerical-methods chapter. Here, just know the two shapes well. (9709/32 May/Jun 2022 Q5(a))

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