Power, Exponential and Logarithmic Functions

Try this

Drag $P$ . How are the coordinates of $P$ related to those of $Q$ ? How does the segment $P Q$ sit relative to the line $y = x$ ?
Lower $a$ from $2$ . At roughly what value of $a$ do the two curves start to have common points? (Try $a = 1.44$ .)
Go to $0 < a < 1$ , say $a = 0.5$ . Where are the common points? Then try $a = 0.0625$ .
Switch to “Power functions” and click through the values of $α$ . Which point do all the graphs pass through? How does the monotonicity on $(0, + \infty)$ differ between $α > 0$ and $α < 0$ ?

$y = a^{x}$ ( $a > 0$ , $a \neq = 1$ ) has domain $R$ and range $(0, + \infty)$ ; its inverse is $y = lo g_{a} x$ , with domain $(0, + \infty)$ and range $R$ . Both are increasing when $a > 1$ and both decreasing when $0 < a < 1$ . The graphs of two mutually inverse functions are mirror images in the line $y = x$ :

(t, a^{t}) lies on y = a^{x} ⟺ (a^{t}, t) lies on y = lo g_{a} x .

The graph of every power function $y = x^{α}$ ( $α$ a constant) passes through $(1, 1)$ ; on $(0, + \infty)$ it is increasing if $α > 0$ and decreasing if $α < 0$ . For $α = \frac{p}{q}$ (in lowest terms, $q > 0$ ): if $q$ is even, the domain contains no negative numbers; if $q$ is odd, the function is even when $p$ is even and odd when $p$ is odd.

Click all eight values of

α

and use the three readings on the right to check the paragraph above.

Predict first

How many common points do the graphs of $y = 1. 2^{x}$ and $y = lo g_{1.2} x$ have?

0

1

2

3

Choose an answer first; then you can see the result.

Why: for a > 1 the common points all lie on the mirror

For a > 1, every common point of y = aˣ and y = logₐx lies on the line y = x

Step 1 / 5

The mirror relation
$P$ is on the exponential curve; swapping its coordinates gives $Q$ on the log curve, and the line $y = x$ is the perpendicular bisector of $P Q$ .

$y = a^{x} ⟺ x = lo g_{a} y$
Name a common point
Use monotonicity
A bigger base never touches
The proof breaks for 0 < a < 1

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: every common point of $y = a^{x}$ and $y = lo g_{a} x$ lies on the line $y = x$

$a = 1.2$ : both common points land on the dashed line $y = x$ .

Practice

Exercise 1Basic

Solve $2^{x} = 8^{x - 1}$ .

Exercise 2Basic

Find the maximum of $y = lo g_{\frac{1}{2}} (x^{2} - 4 x + 5)$ .

Exercise 3Advanced

The interval on which $y = lo g_{\frac{1}{2}} (x^{2} - 2 x - 3)$ is increasing is $(- \infty, t)$ . Find $t$ .

Exercise 4Challenge

How many real solutions does $(\frac{1}{16})^{x} = lo g_{\frac{1}{16}} x$ have?

Try this

Why: for a > 1 the common points all lie on the mirror

For a > 1, every common point of y = aˣ and y = logₐx lies on the line y = x

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Finding where a composite log function increases, forgetting the domain