Logarithmic Function Manipulation

Part 1 · Hook

Why this matters

Before calculators, engineers multiplied big numbers by adding. They looked up the logarithms of the two numbers, added them, and looked up which number had that sum as its logarithm. That trick works because a logarithm turns multiplication into addition, division into subtraction and powers into multiplication.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which equals $2^3 \cdot 2^5$ ?

$2^8$
$2^{15}$
$4^8$
$4^{15}$

Show the answer

Multiplying powers of one base adds the exponents.

Correct: $2^8$ :
$2^{15}$ :
$4^8$ :
$4^{15}$ :

2. What is $\log_3 27$ ?

$3$
$9$
$24$
$\frac{1}{3}$

Show the answer

$3^3 = 27$ .

Correct: $3$ :
$9$ :
$24$ :
$\frac{1}{3}$ :

3. Multiplying the input of $\log x$ by $100$ does what to the output?

Adds $2$
Multiplies it by $2$
Adds $100$
Multiplies it by $100$

Show the answer

$100 = 10^2$ , so the output increases by $2$ .

Correct: Adds $2$ :
Multiplies it by $2$ :
Adds $100$ :
Multiplies it by $100$ :

Part 4 · See it

See it first

Graphs of f(x) = log base 2 of x and h(x) = log base 2 of 4x; h is f moved up 2 units, so (1, 0) becomes (1, 2) and (4, 2) becomes (4, 4). — $h(x)$ $= \log_2(4x)$ $= 2 + \log_2 x$ : multiplying the input by $4$ is the same as shifting the graph of $\log_2 x$ up $2$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Exponents add when powers multiply: $b^m b^n = b^{m + n}$ product property: $\log_b(xy)$ $= \log_b x + \log_b y$
Exponents subtract when powers dividequotient property: $\log_b \frac{x}{y}$ $= \log_b x - \log_b y$
Exponents multiply when a power is raised to a powerpower property: $\log_b(x^n) = n\log_b x$
Rewrite $\log_b x$ through another base $a$ change of base: $\log_b x$ $= \frac{\log_a x}{\log_a b}$ , so every logarithm is a vertical dilation of every other

Part 6 · Key ideas

Key ideas

$\log_b(xy)$ $= \log_b x + \log_b y$ , $\log_b \frac{x}{y}$ $= \log_b x - \log_b y$ , $\log_b(x^n) = n\log_b x$ , for $x, y > 0$ .
Change of base: $\log_b x = \frac{\ln x}{\ln b}$ . So $\log_4 x = \frac{1}{2}\log_2 x$ , a vertical dilation.
Graphs: $\log_b(kx)$ $= \log_b k + \log_b x$ is a vertical shift (and a horizontal dilation); $\log_b(x^n) = n\log_b x$ is a vertical dilation.
Watch the domain: $\ln(x^2)$ is defined for $x \ne 0$ , but $2\ln x$ only for $x > 0$ .

Part 7 · Misconception

A common mistake

The wrong idea: $\log(x + y)$ $= \log x + \log y$ , so $\log 2 + \log 3 = \log 5$ .

What actually happens: The logarithm of a product is a sum: $\log 2 + \log 3 = \log 6$ . There is no rule that splits $\log(x + y)$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graphs of $f$ and $g$

The graphs show

f(x) = \log_3 x

and

g(x) = \log_3(9x)

.

1. For every $x > 0$ , what is $g(x) - f(x)$ ?

$2$
$9$
$3$
$8x$

Show the answer

By the product property, $\log_3(9x)$ $= \log_3 9$ $+ \log_3 x$ $= 2$ $+ \log_3 x$ . So $g(x) - f(x) = 2$ , as the graph shows at $x = 1$ , $3$ and $9$ .

Correct: $2$ : $\log_3 9 = 2$ , the same gap at every $x$ .
$9$ : The factor $9$ inside the logarithm becomes $\log_3 9 = 2$ outside it.
$3$ : $\log_3 3 = 1$ ; the factor is $9$ , and $\log_3 9 = 2$ .
$8x$ : Subtracting the inputs is not how the outputs compare; the gap is a constant.

2. Which transformation of the graph of $f$ gives the graph of $g$ ?

A vertical translation up $2$ units
A horizontal translation left $9$ units
A vertical dilation by a factor of $9$
A vertical translation up $9$ units

Show the answer

$g(x) = 2 + f(x)$ : every output of $f$ goes up by $2$ .

Correct: A vertical translation up $2$ units: $g(x) = f(x) + 2$ .
A horizontal translation left $9$ units: A shift left would be $\log_3(x + 9)$ ; here $x$ is multiplied by $9$ .
A vertical dilation by a factor of $9$ : A vertical dilation multiplies outputs; $g(1) = 2$ while $9 f(1) = 0$ .
A vertical translation up $9$ units: The shift is $\log_3 9 = 2$ , not $9$ .

3. The graph of $g$ can also be described as which transformation of the graph of $f$ ?

A horizontal dilation by a factor of $\frac{1}{9}$
A horizontal dilation by a factor of $9$
A vertical dilation by a factor of $2$
A horizontal translation right $2$ units

Show the answer

$g(x) = f(9x)$ : $g$ reaches each output of $f$ at an input $\frac{1}{9}$ as large, so the graph is squeezed toward the $y$ -axis. For a logarithm, that horizontal dilation is the same as a vertical shift.

Correct: A horizontal dilation by a factor of $\frac{1}{9}$ : $f(9) = 2$ and $g(1) = 2$ : each input is divided by $9$ .
A horizontal dilation by a factor of $9$ : Multiplying the input by $9$ squeezes the graph; a factor of $9$ would stretch it.
A vertical dilation by a factor of $2$ : $2f(x) = \log_3 x^2$ , which does not match $g(1) = 2$ .
A horizontal translation right $2$ units: A shift right would lower the graph; $g$ is above $f$ .

Table

Values of $\ln x$

Use the table and the properties of logarithms, without a calculator.

Natural logarithms of 2, 3, 5 and 7, rounded to three decimal places
$x$	2	3	5	7
$\ln x$	0.693	1.099	1.609	1.946

4. Which value is closest to $\ln 6$ ?

$1.79$
$0.76$
$2.08$
$0.41$

Show the answer

$\ln 6$ $= \ln(2 \cdot 3)$ $= \ln 2$ $+ \ln 3$ $\approx 0.693$ $+ 1.099$ $= 1.792$ .

Correct: $1.79$ : Product property: add the logarithms of the factors.
$0.76$ : That multiplies $\ln 2$ by $\ln 3$ ; the logarithm of a product is a sum.
$2.08$ : That is about $\ln 8 = 3\ln 2$ .
$0.41$ : That is $\ln 3 - \ln 2 = \ln 1.5$ .

5. Which value is closest to $\ln 25$ ?

$3.22$
$2.59$
$8.05$
$0.80$

Show the answer

$\ln 25$ $= \ln(5^2)$ $= 2\ln 5$ $\approx 2(1.609)$ $= 3.218$ .

Correct: $3.22$ : Power property: the exponent $2$ multiplies $\ln 5$ .
$2.59$ : That is $(\ln 5)^2$ ; the power property gives $2\ln 5$ , not a square.
$8.05$ : That is $5\ln 5$ ; $25 = 5^2$ , so the factor is $2$ .
$0.80$ : That is half of $\ln 5$ , which would be $\ln \sqrt{5}$ .

6. Write $\ln(x^4) - \ln(x^2)$ as a single logarithm that is equal to it for every $x \ne 0$ .

Type an expression for $\ln(x^4) - \ln(x^2) =$ .

Show the answer

Quotient property: $\ln(x^4) - \ln(x^2)$ $= \ln \frac{x^4}{x^2}$ $= \ln(x^2)$ . Both sides are defined for every $x \ne 0$ .

Answer: $\ln(x^4) - \ln(x^2) =$ $\ln(x^{2})$

7. Which equation is true for every pair of positive numbers $x$ and $y$ ?

$\log(xy) = \log x + \log y$
$\log(x + y)$ $= \log x + \log y$
$\log \frac{x}{y}$ $= \frac{\log x}{\log y}$
$(\log x)^2 = 2\log x$

Show the answer

The product property turns a product inside a logarithm into a sum outside it. There is no rule for a sum inside.

Correct: $\log(xy) = \log x + \log y$ : The product property.
$\log(x + y)$ $= \log x + \log y$ : Try $x = y = 1$ : $\log 2 \ne 0$ .
$\log \frac{x}{y}$ $= \frac{\log x}{\log y}$ : The quotient property gives $\log x - \log y$ .
$(\log x)^2 = 2\log x$ : The power property is $\log(x^2) = 2\log x$ ; $(\log x)^2$ is a square of the logarithm.

Part 9 · Summary

Summary

Logarithms turn products into sums, quotients into differences and powers into multiples, because exponents behave that way. Change of base writes any logarithm as a multiple of another, so every logarithmic function is a vertical dilation of every other. On graphs, multiplying the input shifts the graph up or down. Each rule needs positive inputs; watch for domain changes such as

\ln(x^2)

versus

2\ln x

.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graphs of ff and gg

Values of ln⁡x\ln x

Summary

What comes next

Connections

The graphs of $f$ and $g$

Values of $\ln x$