Grade by Grade
A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Logarithms: The Exponent You Are Looking For

A1.1About 70 minutes
Learning goals
  • I can explain that a logarithm is an exponent: log_b x is the power of b that gives x.
  • I can describe the logarithm as the inverse of exponentiation (the operation that undoes it).
  • I can evaluate simple logarithms such as log₂ 32, log₃ (1/9) and log₄ 8 without a calculator.
  • I can explain why log_b 1 = 0, log_b b = 1 and why the logarithm of 0 or a negative number is undefined.

Words to know

logarithm
log_b x is the exponent you put on the base b to get x. So log_b x = y means exactly the same as bʸ = x.
base
The number b that is raised to a power. For logarithms the base must be positive and not equal to 1.
argument
The number x you take the logarithm of, as in log_b x. It must be positive.
inverse operation
An operation that undoes another one. Taking log₂ undoes raising 2 to a power, just as a square root undoes squaring a positive number.
common logarithm
A logarithm with base 10, usually written log x with no base shown.

A question your calculator can't ask directly

You already know how to answer "What is 2⁵?" You multiply: 2 × 2 × 2 × 2 × 2 = 32.

Now turn the question around: "2 to what power gives 32?" You know the answer is 5. But what about "2 to what power gives 50?" or "1.03 to what power gives 2?" That second question is exactly what a bank asks when it works out how long it takes money to double at 3% interest a year.

Questions like these ask for an exponent. Mathematicians gave that missing exponent a name: the logarithm. In this unit you will learn to evaluate logarithms, graph them, use their laws and solve equations where the unknown sits in an exponent. These are skills you will use again in Calculus and Vectors (MCV4U), in first-year university science and engineering, and every time you see a pH value or a decibel reading.

Remember from Grade 11: in Functions (MCR3U) you graphed f(x) = bˣ, used the exponent laws with integer and rational exponents (for example 8^(2/3) = 4), and learned that the inverse of a function undoes it, with the graph reflected in the line y = x. You also met the hint that the inverse of an exponential function is called the logarithmic function. Now we meet it properly.

The definition

For a base b > 0, b ≠ 1, and a number x > 0:

log_b x = y means bʸ = x.

Read log_b x as "log base b of x". In words: the logarithm is the exponent. The two equations are just two ways of writing the same fact.

  • 2⁵ = 32, so log₂ 32 = 5.
  • 10³ = 1000, so log₁₀ 1000 = 3.
  • 3⁻² = 1/9, so log₃ (1/9) = −2.

Notice what lives where. In both forms the base is the same number. In exponential form the answer to the power is the result (32). In logarithmic form that result moves inside the log as the argument, and the logarithm's value is the exponent (5).

exponential formlogarithmic form25= 32log232= 5same factbaseexponentresultbaseresultexponent
2⁵ = 32 and log₂ 32 = 5 say the same thing. The base stays the base; the exponent becomes the value of the logarithm.

Say it as a question

Whenever you see log_b x, say to yourself: "b to what power gives x?" For log₃ 81, ask "3 to what power gives 81?" Since 3⁴ = 81, log₃ 81 = 4.

Worked example

Example 1: Evaluating logarithms of whole numbers

Evaluate (a) log₂ 64, (b) log₅ 125, (c) log₁₀ 100 000.

  1. (a) 2 to what power gives 64? 2⁶ = 64, so log₂ 64 = 6.
  2. (b) 5 to what power gives 125? 5³ = 125, so log₅ 125 = 3.
  3. (c) 10 to what power gives 100 000? Count the zeros: 10⁵ = 100 000, so log₁₀ 100 000 = 5.

(a) 6 (b) 3 (c) 5

Fractions and negative logarithms

A logarithm can be negative. This happens when the argument is between 0 and 1 (for a base bigger than 1). Remember from Grade 11 that b⁻ⁿ = 1/bⁿ.

Worked example

Example 2: Arguments that are fractions

Evaluate (a) log₂ (1/8), (b) log₃ (1/81), (c) log₁₀ 0.01.

  1. (a) 1/8 = 1/2³ = 2⁻³. So log₂ (1/8) = −3.
  2. (b) 1/81 = 1/3⁴ = 3⁻⁴. So log₃ (1/81) = −4.
  3. (c) 0.01 = 1/100 = 10⁻². So log₁₀ 0.01 = −2.

(a) −3 (b) −4 (c) −2. A negative logarithm does not mean a negative argument. It means the argument is a fraction between 0 and 1.

Fractional logarithms

A logarithm can also be a fraction. Use rational exponents: b^(1/n) is the n-th root of b, and b^(m/n) is the n-th root of b, raised to the power m.

Worked example

Example 3: Rewrite both numbers with the same base

Evaluate (a) log₄ 2, (b) log₉ 27, (c) log₈ (1/4).

  1. (a) 2 = √4 = 4^(1/2). So log₄ 2 = 1/2.
  2. (b) 9 and 27 are both powers of 3. Write 27 as a power of 9: 9^(3/2) = (√9)³ = 3³ = 27. So log₉ 27 = 3/2.
  3. Another way for (b): let log₉ 27 = y. Then 9ʸ = 27, so (3²)ʸ = 3³, which means 3^(2y) = 3³. Then 2y = 3 and y = 3/2.
  4. (c) Let log₈ (1/4) = y. Then 8ʸ = 1/4, so (2³)ʸ = 2⁻², which means 3y = −2 and y = −2/3.

(a) 1/2 (b) 3/2 (c) −2/3

xlog₂ x1/8−31/4−21/2−110214283164325
Each step to the right on the top line multiplies x by 2. Each step on the bottom line adds 1 to log₂ x. Logarithms turn multiplication into addition.

Special values every log has

For any allowed base b:

  • log_b 1 = 0, because b⁰ = 1.
  • log_b b = 1, because b¹ = b.
  • log_b bⁿ = n, because the exponent on b is n. (For example, log₇ 7¹² = 12.)
  • b^(log_b x) = x for x > 0. Raising b to "the power of b that gives x" gives x.

The last two facts show that "raise b to a power" and "take log_b" undo each other. That is what it means for the logarithm to be the inverse of exponentiation. In lesson 7 (Exponential and Logarithmic Functions as Inverses) you will see this on a graph.

What you cannot take the log of

With a positive base b, every power bʸ is positive. No power of 2 equals 0, and no power of 2 equals −8. So:

  • log_b 0 is undefined, and
  • log_b of a negative number is undefined (in the real numbers).

The base also has rules. Base 1 is useless (1ʸ = 1 always, so log₁ 5 would have no answer), and negative or zero bases cause trouble with fractional exponents. So we always use b > 0 and b ≠ 1.

Exponential formLogarithmic formQuestion it answers
3⁴ = 81log₃ 81 = 43 to what power is 81?
10⁻¹ = 0.1log₁₀ 0.1 = −110 to what power is 0.1?
16^(1/4) = 2log₁₆ 2 = 1/416 to what power is 2?
7⁰ = 1log₇ 1 = 07 to what power is 1?
(1/2)³ = 1/8log_(1/2) (1/8) = 31/2 to what power is 1/8?
Each row shows one fact written two ways.
Worked example

Example 4: Using the inverse property

Evaluate (a) log₆ 6⁹, (b) 5^(log₅ 13), (c) log₂ (log₃ 9).

  1. (a) The exponent on 6 is 9, so log₆ 6⁹ = 9.
  2. (b) log₅ 13 is the power of 5 that gives 13. Raising 5 to that power gives 13.
  3. (c) Work from the inside out. log₃ 9 = 2 because 3² = 9. Then log₂ 2 = 1.

(a) 9 (b) 13 (c) 1

Just for fun

The word logarithm was built from two Greek words, logos (often translated as "ratio") and arithmos ("number"). So a logarithm is a "ratio number"... which is a fancy way of saying "the exponent you were looking for all along."

Hands-on activity

Fold your way to a logarithm

You need: one sheet of paper (newspaper or printer paper), a pencil and a notebook.

  1. Fold the sheet in half. Count the layers (2). Fold again (4 layers). Keep going as long as you can. Most people manage 6 or 7 folds.
  2. Make a table: number of folds n, and number of layers L. You should see L = 2ⁿ.
  3. Now turn it around. Beside each number of layers, write log₂ L. What do you notice? (It equals the number of folds.)
  4. Without folding: how many folds would give 1024 layers? How many would give 1 layer? Explain using logarithms.
  5. A sheet of printer paper is about 0.1 mm thick. If you could keep folding, how many folds would make a stack at least 1 m (1000 mm) thick? (Hint: you need 0.1 × 2ⁿ ≥ 1000, so 2ⁿ ≥ 10 000. Use powers of 2 to find the smallest n.)

What to look for: log₂ L always equals the number of folds. 1024 = 2¹⁰ needs 10 folds; 1 layer needs 0 folds (log₂ 1 = 0). For step 5, 2¹³ = 8192 is too small and 2¹⁴ = 16 384 is enough, so 14 folds. The number of layers grows fast, but the logarithm (the number of folds) grows slowly.

Your turn

1

Evaluate log₃ 27.

2

Evaluate log₂ (1/16).

3

Evaluate log₁₀ 0.001.

4

Evaluate log₂₅ 5.

5

Evaluate log₄ 32.

6

Evaluate log₁₂ 1 and log₁₂ 12.

7

Evaluate 3^(log₃ 20).

8

Explain why log₅ (−25) is undefined.

More practice: warm-up

1

Evaluate log₂ 8.

2

Evaluate log₅ 25.

3

Evaluate log₁₀ 10 000.

4

Evaluate log₇ 7.

5

Evaluate log₉ 1.

More practice: on your own

1

Evaluate log₂ (1/32) and log_(1/2) 32. Compare them.

2

Evaluate log₂₇ 9.

3

Evaluate log₃ √3 and log₃ (1/√3).

4

Evaluate log₁₀ (log₂ 1024).

5

Between which two whole numbers is log₃ 50? Explain.

Challenge

1

Evaluate log₈ 4 + log₄ 8.

2

Find every whole-number base b > 1 for which log_b 64 is a whole number.

3

Show that log₂ 3 cannot be a rational number p/q (with p and q positive whole numbers).

Check your understanding

Quiz

1.

What is the value of log₂ 16?

2.

Which equation means the same as log₅ x = 3?

3.

What is log₃ (1/27)?

4.

What is log₄ 2?

5.

Which of these is undefined?

6.

What is 7^(log₇ 40)?

For parents and teachers

Big idea (A1.1): a logarithm is an exponent. log_b x answers "b to what power gives x?" Everything in this unit grows from that single sentence.

How to help: Name a power fact ("10 cubed is 1000") and ask your teen to say it in log form ("log base 10 of 1000 is 3"). Then do the reverse. Five minutes of this back-and-forth builds fluency fast.

Common difficulties:

  • Mixing up where the numbers go: in log₂ 32 = 5, the 5 is the exponent, not the result.
  • Thinking a negative logarithm means a negative argument. log₂ (1/8) = −3 has a positive argument.
  • Forgetting that log_b 1 = 0 (not 1) and that log_b 0 is undefined.
  • Not rewriting both numbers with a common base in questions like log₉ 27.