Unit 2 · Topic 2.9 · about 30 minutes · 7 parts · 14 scored checks, 2 of them multiple choice

Logarithmic Expressions

Built from College Board's course framework · Every answer key independently checked

The graph of y = 2 to the x. A dashed line at height 8 meets the curve at x = 3, so log base 2 of 8 is 3. A dashed line at height 20 meets the curve at about x = 4.32, between the points (4, 16) and (5, 32), so log base 2 of 20 is between 4 and 5.

Part 1 Think first

Folding a sheet of paper

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

Fold a sheet of paper in half and it is 2 layers thick. Fold it again and it is 4 layers, then 8. Suppose you could keep folding as long as you liked. About how many folds would it take to get past 1,000 layers?

In short. The logarithm $\log_{b} c$ is the exponent you raise the base $b$ to in order to get $c$ .

$\log_{b} c = a$ says exactly what $b^{a} = c$ says, for $b > 0$ and $b \neq 1$ . With no base written, the base is 10, the common logarithm.
When $c$ is a power of $b$ , arithmetic gives $\log_{b} c$ exactly. Otherwise, find the two powers of $b$ that $c$ sits between, and use technology for a decimal.
On a logarithmic scale each unit multiplies by the base, so base 10 marks read 1, 10, 100, 1000, evenly spaced.

Part 2 Learn it

Four short ideas, each checked as you go

2a. A logarithm is an exponent

A logarithm answers one question: what exponent turns the base into this number? $\log_{2} 8$ asks what power of 2 gives 8. Since $2^{3} = 8$ , $\log_{2} 8 = 3$ .

In symbols:

\log_{b} c = a exactly when b^{a} = c

The base $b$ must be positive and not equal to 1. (Every power of 1 is 1, so base 1 could never reach 8.) The value $a$ is an exponent, so it can be negative or a fraction.

It helps to read a logarithm as a count of factors. $\log_{3} 81 = 4$ because $81 = 3 \cdot 3 \cdot 3 \cdot 3$ , four factors of 3.

When no base is written, the base is 10. That is the common logarithm: $\log 1000 = 3$ because $10^{3} = 1000$ .

To switch forms, keep the base as the base. In $\log_{5} 25 = 2$ the base is 5 and the exponent is 2, so the exponential form is $5^{2} = 25$ . Writing $2^{5}$ swaps the two, and $2^{5}$ is 32.

Quick check · Match

No calculator. Tap a logarithm, then its value.

2b. Values you can find by hand

When $c$ is a power of the base, $\log_{b} c$ comes straight from arithmetic. Write $c$ as a power of $b$ and read off the exponent. The exponent properties from 2.4 Exponential Function Manipulation handle the cases that look strange.

$\log_{b} 1 = 0$ , since $b^{0} = 1$ for every allowed base.
$\log_{b} b = 1$ , since $b^{1} = b$ .
A reciprocal gives a negative value: $\log_{2} \frac{1}{8} = - 3$ , because $2^{- 3} = \frac{1}{8}$ .
A square root gives one half: $\log_{7} \sqrt{7} = \frac{1}{2}$ , because $7^{\frac{1}{2}} = \sqrt{7}$ .

Common logarithms of powers of 10 count places. $\log 100000 = 5$ , and $\log 0.01 = - 2$ because $0.01 = \frac{1}{100} = 10^{- 2}$ .

A negative logarithm does not mean a negative number. $\log 0.01 = - 2$ says 0.01 is two factors of 10 below 1. The number 0.01 is still positive, like every power of a positive base.

Quick check · Fill the statement

No calculator. Find $\log_{5} \frac{1}{125}$ . Tap a blank, then tap a word or a value.

$\log_{5} \frac{1}{125}$ asks what power of gives $\frac{1}{125}$ . A exponent gives a reciprocal, so $\log_{5} \frac{1}{125}$ is .

2c. Between two powers

Most numbers are not powers of the base. $\log_{3} 50$ asks what power of 3 gives 50, and no whole number works: $3^{3} = 27$ and $3^{4} = 81$ . Since 50 is between 27 and 81, $\log_{3} 50$ is between 3 and 4.

You can always get that bracket by hand. List powers of the base until you pass the number. It works below 1 as well: $\log 0.5$ is between $- 1$ and 0, because $10^{- 1} = 0.1$ and $10^{0} = 1$ .

For a decimal, use technology. On most graphing calculators the LOG key is the common logarithm, and many also have a template for any base. A calculator gives $\log_{3} 50 \approx 3.56$ .

Look at where that lands. 50 is below 54, the number halfway between 27 and 81, yet $\log_{3} 50$ is above 3.5. Halfway in the exponent means multiplying 27 by $3^{\frac{1}{2}} \approx 1.73$ , which gives about 46.8, not 54. Equal steps in a logarithm are equal multiplications, so you cannot place a logarithm by measuring evenly between two powers.

Quick check · Sort it

No calculator. Tap a logarithm, then the two integers its value is between.

between 1 and 2

between 2 and 3

between 3 and 4

2d. Logarithmic scales

A ruler adds. Its marks are 0, 1, 2, 3, and each unit is one more than the last. A logarithmic scale multiplies instead. On a base 10 logarithmic scale the evenly spaced marks are $10^{0}, 10^{1}, 10^{2}, 10^{3}$ , which is 1, 10, 100, 1000. Each unit to the right multiplies by 10.

A number's position on that scale is its common logarithm. 1000 sits at 3, because $\log 1000 = 3$ . Two units to the right of 1000 is $1000 \cdot 10^{2} = 100000$ , not 1002.

This is why a logarithmic scale can show a village of 300 people and a city of 3 million on one line, with both easy to read. Distance on the scale measures a ratio. Numbers one unit apart differ by a factor of 10, whatever their size, so 3 and 30 are exactly as far apart as 30,000 and 300,000.

A base 2 scale works the same way with doubling: the marks are 1, 2, 4, 8, 16, and each unit is a factor of 2.

Quick check · Multiple choice

No calculator. On a base 10 logarithmic scale, point P is 3 units to the right of point Q. How does the number at P compare with the number at Q?

Part 3 Watch it done

An exact value that is a fraction

Watch each step, then do the last one yourself.

Without a calculator, find the exact value of $\log_{8} 4$ .

Step 1 · Turn it into an exponent question
Call the value $a$ . Then $\log_{8} 4 = a$ means $8^{a} = 4$ . No whole number works, since $8^{0} = 1$ and $8^{1} = 8$ , so $a$ is between 0 and 1.
Step 2 · Write both numbers with one base
8 and 4 are both powers of 2: $8 = 2^{3}$ and $4 = 2^{2}$ . By the power property, $8^{a} = (2^{3})^{a} = 2^{3 a}$ , so the equation becomes $2^{3 a} = 2^{2}$ .
Why this step?
Why does $2^{3 a} = 2^{2}$ mean $3 a = 2$ ?
The function $2^{x}$ is always increasing, so no two different exponents give the same power of 2. If $2^{3 a}$ and $2^{2}$ are the same number, the exponents $3 a$ and 2 must be the same too.
Your step · Calculate

No calculator. Finish it. What is $\log_{8} 4$ ? Give a decimal rounded to three places.

Your answer

Part 4 Try it

When the value is not exact

A town of 4,700 people goes on a base 10 logarithmic scale at the position $\log 4700$ . That is not a whole number, so this time you bracket it by hand and then let the calculator finish.

Try it · Calculate

Graphing calculator required. First decide which two integers $\log 4700$ is between. Then use your calculator to find $\log 4700$ , rounded to three decimal places.

Your answer

Practice

Practice until it is automatic

Exact values, new every time. Write the number as a power of the base and read off the exponent. Watch for a reciprocal, which gives a negative value, a square root, which gives one half, and 1, which gives 0.

Logarithm values practice page

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page. Check your answer against the model and each scoring point.

Free response is 37.5% of the exam score. This one is shaped like Question 4, Symbolic Manipulations.

Free-response practice · Question 4 style

No calculator. (A) Write $\log_{9} \frac{1}{3} = a$ in exponential form, and find the exact value of $a$ . (B) The value of $\log_{2} 50$ is not an integer. Find the two consecutive integers that $\log_{2} 50$ is between, and justify your answer.

Your answer stays on this page. It is never sent or saved.

Model answer

(A) In exponential form, $9^{a} = \frac{1}{3}$ . Since $9 = 3^{2}$ and $\frac{1}{3} = 3^{- 1}$ , this is $3^{2 a} = 3^{- 1}$ , so $2 a = - 1$ and $a = - \frac{1}{2}$ . Check: $9^{- \frac{1}{2}} = \frac{1}{\sqrt{9}} = \frac{1}{3}$ . (B) $\log_{2} 50$ is between 5 and 6. The value of $\log_{2} 50$ is the exponent that 2 must be raised to in order to get 50. Since $2^{5} = 32$ and $2^{6} = 64$ , and $32 < 50 < 64$ , that exponent is between 5 and 6.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes the exponential form $9^{a} = \frac{1}{3}$ , with 9 as the base.
Finds $a = - \frac{1}{2}$ , by writing both sides as powers of 3 or by another correct method.
States that $\log_{2} 50$ is between 5 and 6.
Justifies the interval with $2^{5} = 32$ and $2^{6} = 64$ , and the fact that 50 is between them.

Part 6 Check yourself

Bacteria counts on a logarithmic chart

The three case questions use the same table. Answer them from it.

A lab counts the bacteria in five samples. Its chart uses a base 10 logarithmic scale and puts each sample at the common logarithm of its count. For example, a count of 1,000 sits at the mark 3.

Bacteria counts in five samples
Sample	Count
A	1,000
B	10,000,000
C	300
D	3,000
E	100,000

Case question 1 · Calculate

No calculator. How many units apart are samples A and B on the chart?

Your answer

Case question 2 · Multiple choice

No calculator. Which two samples are exactly one unit apart on the chart?

Case question 3 · Calculate

No calculator. A sixth sample sits exactly 2 units to the left of sample A on the chart. What is its count?

Your answer

The rest stand on their own and do not use the case.

Check yourself · Put it in order

No calculator. Put the values in order from least to greatest.

Check yourself · Spot the error

No calculator. A student estimates $\log_{2} 40$ . Tap the first line with a mistake.

Check yourself · Decide and justify

No calculator. A classmate says $\log_{2} 8$ and $\log_{8} 2$ are equal, because both use the numbers 2 and 8. Is he right?

Which reason supports your answer?

Part 7 Keep it

Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 2.4 Exponential Function Manipulation · Fill the statement

No calculator. Find $32^{- \frac{1}{5}}$ . Tap a blank, then tap a word or a value.

The $\frac{1}{5}$ takes the of 32. The negative sign takes the of that, so $32^{- \frac{1}{5}}$ is .

From 2.1 Change in Arithmetic and Geometric Sequences · Fill the statement

No calculator. A patient takes 400 milligrams of a medicine. The amounts left in the body, measured every 6 hours, are 400, 200, 100, 50 milligrams. Let $g_{n}$ be the amount left after $n$ six-hour periods. Tap a blank, then tap a word.

Each term is the term before, so the sequence is with common ratio .

In three lines

$\log_{b} c = a$ means $b^{a} = c$ : the logarithm is the exponent, and with no base written the base is 10.
Exact values come from writing $c$ as a power of $b$ ; otherwise bracket $c$ between two powers and use technology for a decimal.
On a logarithmic scale each unit multiplies by the base, so distance measures a ratio.

Vocabulary

Logarithmic expression: $\log_{b} c$ , the exponent that the base $b$ must be raised to in order to get $c$ ; $\log_{b} c = a$ exactly when $b^{a} = c$ .
Base: The number $b$ in $\log_{b} c$ , which must satisfy $b > 0$ and $b \neq 1$ .
Common logarithm: The logarithm with base 10, written $\log c$ with no base shown, so $\log 1000 = 3$ .
Logarithmic scale: A scale whose evenly spaced marks are powers of the base, such as $10^{0}, 10^{1}, 10^{2}$ , so each unit is a multiplicative change by the base.

Course alignment, for teachers

AP Precalculus topic 2.9, Unit 2: Exponential and Logarithmic Functions.

Skill 1.B: Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context.

This lesson maps all 3 essential knowledge statements of the topic to the part that teaches each one: 3 are checked by a scored question. The map is in the lesson file.