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Active Lesson: Logarithmic Functions

Mathimatikos TeamJune 8, 20269 min read

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Before You Start

Exponential functions
Equation solving
Calculator use for powers

Lesson time

20 min

Objectives

What You Will Learn

Explain a logarithm as the answer to an exponent question.
Use logarithm laws to rewrite expressions.
Solve exponential and logarithmic equations while checking domains.

Activity

Preview Activity: How Long Until Half Remains?

A battery starts at $80%$ and loses half of its remaining charge every $6$ hours. A simple model is $B (t) = 80 (0.5)^{t /6}$ .

Find $B (0)$ , $B (6)$ , and $B (12)$ .
Estimate when the battery reaches $20%$ .
Explain why solving $20 = 80 (0.5)^{t /6}$ requires undoing an exponent.

Hint

Divide both sides by $80$ first. Then ask: what exponent makes $0.5$ equal to $0.25$ ?

Solution

$B (0) = 80$ , $B (6) = 40$ , and $B (12) = 20$ , so the battery reaches $20%$ after $12$ hours.

Definition

The Logarithm

lo g_{b} x = y if and only if b^{y} = x

The logarithm $lo g_{b} x$ asks: to what power must $b$ be raised to get $x$ ?

Because logarithms undo exponentials, they are the natural tool when the unknown is in the exponent.

Exponential and Logarithm as Inverses

The graphs of $y = e^{x}$ and $y = ln x$ mirror each other across $y = x$ .

Interactive Graph

Definition

Core Logarithm Laws

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$ turns multiplication into addition.

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$ turns division into subtraction.

$lo g_{b} (x^{r}) = r lo g_{b} x$ moves powers down as factors.

$ln (e^{x}) = x$ and $e^{l n x} = x$ show the inverse relationship.

Worked example

Worked Example: Solve an Exponential Equation

Solve $3^{x} = 17$ .

Solution

Take natural logs on both sides: $ln (3^{x}) = ln (17)$ .

Use the power law: $x ln 3 = ln 17$ .

Therefore $x = \frac{l n 17}{l n 3} \approx 2.58$ .

Activity

Activity: Expand and Condense

Expand $lo g (x^{2} y^{5})$ .
Condense $ln 4 + 3 ln x - ln (y + 1)$ into one logarithm.
Solve $ln (x - 2) = ln 7$ and state the domain restriction.

Hint

Use the product, quotient, and power laws. For the last question, $x - 2$ must be positive.

Solution

$lo g (x^{2} y^{5}) = 2 lo g x + 5 lo g y$ .

$ln 4 + 3 ln x - ln (y + 1) = ln (\frac{4 x ^{3}}{y + 1})$ .

$x - 2 = 7$ , so $x = 9$ , and the original equation requires $x > 2$ .

Summary

A logarithm is an exponent written from the opposite direction.
Log laws are exponent laws translated into logarithm language.
Always check that every logarithm input is positive.

Practice

Try These

Rewrite $2^{5} = 32$ as a logarithmic statement.
Solve $5 e^{0.2 t} = 30$ .
Find the domain of $f (x) = ln (2 x - 6)$ .

Practice logarithms in the Greek quiz

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What You Will Learn

Preview Activity: How Long Until Half Remains?

The Logarithm

Exponential and Logarithm as Inverses

Core Logarithm Laws

Worked Example: Solve an Exponential Equation

Activity: Expand and Condense

Summary

Try These

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