Algebra II

Graphing Exponential and Logarithmic Functions

Lesson

Exponential and logarithmic functions are inverses of each other. Once you know the shape of one, you know the other — just flipped over the line $y = x$ .

Exponential: $y = a \cdot b^x$

$b > 1$ : growth — climbs to the right.
$0 < b < 1$ : decay — falls to the right.
y-intercept is $a$ (because $b^0 = 1$ ).
Horizontal asymptote at $y = 0$ (or wherever a vertical shift moves it). The curve gets close but never touches.

Logarithmic: $y = \log_b(x)$

Vertical asymptote at $x = 0$ (or wherever a horizontal shift moves it).
Passes through $(1, 0)$ when there’s no shift, because $\log_b(1) = 0$ .
Domain: $x > 0$ (logs of zero or negative numbers are undefined).

Worked example 1 — exponential y-intercept

y = 3 \cdot 2^x

At $x = 0$ : $3 \cdot 2^0 = 3 \cdot 1 = 3$ . y-intercept is $3$ .

Worked example 2 — shifted asymptote

y = 2^x + 5

As $x \to -\infty$ , $2^x \to 0$ , so $y \to 5$ . Horizontal asymptote: $y = 5$ .

Worked example 3 — log asymptote

y = \log_3(x - 4)

The asymptote sits where the inside is 0: $x - 4 = 0 \Rightarrow x = 4$ .

Evaluating logs without a calculator

$\log_b(x) = y$ asks: “what power of $b$ gives $x$ ?” Examples:

$\log_2(8) = 3$ because $2^3 = 8$ .
$\log_3(27) = 3$ because $3^3 = 27$ .
$\log_b(1) = 0$ for any base.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{y-intercept of } y = 5^x

Problem 2

\text{y-intercept of } y = 3 \cdot 2^x

Problem 3

\text{Horizontal asymptote of } y = 2^x \text{ (}y\text{ value)}

Problem 4

\text{Vertical asymptote of } y = \log_2(x) \text{ (}x\text{ value)}

Practice

Standard problems matching the lesson.

Problem 5

\text{y-intercept of } y = 4 \cdot 3^x

Problem 6

\text{y-intercept of } y = -2 \cdot 5^x

Problem 7

\text{Horizontal asymptote of } y = 3^x - 5

Problem 8

\text{Horizontal asymptote of } y = 2^x + 8

Problem 9

y = (1/4)^x growth (1) or decay (0)?

Problem 10

y = 7^x growth (1) or decay (0)?

Problem 11

y = (1/3)^x growth (1) or decay (0)?

Problem 12

\text{y-intercept of } y = 2^x

Problem 13

\text{y-intercept of } y = (1/2)^x

Problem 14

\text{Vertical asymptote of } y = \log_3(x - 4)

Problem 15

\text{Vertical asymptote of } y = \log(x + 6)

Problem 16

\log_b(1) = ? \text{ for any base } b

Problem 17

P(t) = 100 * 2^t. Initial population?

Problem 18

N(t) = 80 * (1/2)^t. Initial amount?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\text{Value of } y = 3 \cdot 2^x \text{ at } x = 2

Problem 20

\text{Value of } y = 5^x \text{ at } x = 3

Problem 21

\text{Value of } y = 4 \cdot 2^x \text{ at } x = -1

Problem 22

\text{Value of } y = -3 \cdot 4^x + 7 \text{ at } x = 0

Problem 23

\log_3(27) = ?

Problem 24

\log_2(32) = ?

Problem 25

\log_4(64) = ?

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