A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Common Logarithms and Logarithms to Any Base with Technology

A1.2 A1.1About 70 minutes

Learning goals

I can use the LOG key on a calculator to find common (base-10) logarithms.
I can estimate a logarithm by finding the two whole-number powers it lies between.
I can use the change of base formula log_b x = log x ÷ log b to approximate a logarithm in any base.
I can check a logarithm I calculated by raising the base to that power.

Words to know

common logarithm
A base-10 logarithm. log x with no base written means log₁₀ x.

change of base formula
log_b x = log x ÷ log b. It lets you find a logarithm in any base with the base-10 LOG key.

natural logarithm
A logarithm with base e ≈ 2.718, written ln x. It appears on calculators and is used a lot in calculus; this unit uses base 10 and other bases.

approximation
A rounded decimal value, such as log 2 ≈ 0.301. The exact value is log 2 itself.

Most logs are not whole numbers

In lesson 1 every answer came out "nice": log₂ 32 = 5, log₉ 27 = 3/2. Real problems are messier. If a town's population doubles, how many years at 2% growth does that take? You will need values like log 2 that are not whole numbers or simple fractions.

In 1614 the Scottish mathematician John Napier published the first tables of logarithms, and for more than 300 years scientists, navigators and engineers looked up logarithms in printed tables. Today a scientific calculator does it in a keystroke.

The LOG key: base 10

A logarithm with base 10 is called a common logarithm. It is so common that we usually leave out the base:

log x means log₁₀ x.

Your calculator's LOG key gives common logarithms. Try these and check each one by raising 10 to the answer:

log 1000 = 3, since 10³ = 1000.
log 2 ≈ 0.3010, and 10^0.3010 ≈ 2.000.
log 500 ≈ 2.6990, and 10^2.6990 ≈ 500.
log 0.004 ≈ −2.3979, and 10^(−2.3979) ≈ 0.004.

(On many calculators the 10ˣ key is the second function of LOG. That is no accident: the two operations undo each other.)

On a log scale, equal spaces stand for equal ratios. 1 to 10, 10 to 100 and 100 to 1000 are the same length, and 500 sits at about 2.70 because log 500 ≈ 2.70.

Estimate first

Before you press a key, find the two powers of 10 the number lies between. 500 is between 10² = 100 and 10³ = 1000, so log 500 must be between 2 and 3. If your calculator says 26.99 or 0.27, you know something went wrong.

Worked example

Example 1: Common logarithms and a quick check

Use a calculator to evaluate log 45, log 4500 and log 0.045 to four decimal places. What pattern do you see?

log 45 ≈ 1.6532 (between 1 and 2, since 10 < 45 < 100).
log 4500 ≈ 3.6532 (between 3 and 4).
log 0.045 ≈ −1.3468 (between −2 and −1, since 0.01 < 0.045 < 0.1).
4500 = 45 × 100 and the logarithm went up by exactly 2. Multiplying by 10² adds 2 to the common logarithm. 0.045 = 45 × 10⁻³ and the logarithm went down by 3: 1.6532 − 3 = −1.3468.

log 45 ≈ 1.6532, log 4500 ≈ 3.6532, log 0.045 ≈ −1.3468. Moving the decimal point only changes the whole-number part. In lesson 4 (The Laws of Logarithms) you will see why.

Logarithms in other bases: estimate first

What is log₂ 20? Ask "2 to what power gives 20?" 2⁴ = 16 and 2⁵ = 32. Since 16 < 20 < 32, log₂ 20 is between 4 and 5, and a bit closer to 4.

The line y = 20 meets y = 2ˣ between x = 4 and x = 5. The crossing point is at x = log₂ 20 ≈ 4.32.

The change of base formula

To get a decimal value for log₂ 20 you can use the LOG key with a short argument.

Let y = log₂ 20. Then 2ʸ = 20.

Now write the base 2 as a power of 10: 2 = 10^(log 2). Substitute:

(10^(log 2))ʸ = 20, so 10^(y log 2) = 20.

That says the common logarithm of 20 is y log 2. So log 20 = y log 2, and y = log 20 ÷ log 2.

The same steps work for any base:

log_b x = log x ÷ log b (b > 0, b ≠ 1, x > 0).

So log₂ 20 = log 20 ÷ log 2 ≈ 1.30103 ÷ 0.30103 ≈ 4.3219. Check: 2^4.3219 ≈ 20.00 ✓

Calculator keystrokes

Type it as one expression with brackets: log(20) ÷ log(2). Do not round log 20 and log 2 before you divide; let the calculator carry all the digits. Many newer calculators also have a key such as log_□(□) that lets you type the base directly. Use either one, but know the formula, because you will need it to solve equations.

Worked example

Example 2: Logarithms to other bases

Evaluate to four decimal places, then check: (a) log₃ 20, (b) log₅ 100, (c) log₇ 0.5.

(a) Estimate: 3² = 9 and 3³ = 27, so the answer is between 2 and 3. log₃ 20 = log 20 ÷ log 3 ≈ 2.7268. Check: 3^2.7268 ≈ 20.0 ✓
(b) Estimate: 5² = 25 and 5³ = 125, so between 2 and 3. log₅ 100 = log 100 ÷ log 5 = 2 ÷ 0.69897 ≈ 2.8614. Check: 5^2.8614 ≈ 100.0 ✓
(c) Estimate: 0.5 is between 7⁻¹ ≈ 0.143 and 7⁰ = 1, so the answer is between −1 and 0. log₇ 0.5 = log 0.5 ÷ log 7 ≈ −0.3562. Check: 7^(−0.3562) ≈ 0.500 ✓

(a) ≈ 2.7268 (b) ≈ 2.8614 (c) ≈ −0.3562

Worked example

Example 3: How long to double?

Money in a savings account grows by 3% a year, so each year it is multiplied by 1.03. The number of years n it takes to double satisfies 1.03ⁿ = 2. Find n.

By definition, n = log₁.₀₃ 2.
Change of base: n = log 2 ÷ log 1.03.
log 2 ≈ 0.301030 and log 1.03 ≈ 0.012837, so n ≈ 23.45.

About 23.4 years. (Interest is usually credited once a year, so in practice the balance first passes double after 24 years.) In lesson 15 (Growth and Decay Problems) you will solve many problems like this.

A word about ln

Your calculator also has an ln key. It means the natural logarithm, log base e, where e ≈ 2.718 is a special number that appears naturally in growth and in calculus. You can use ln instead of log in the change of base formula: log₂ 20 = ln 20 ÷ ln 2 ≈ 4.3219 too, because the formula works with any common base on top and bottom. You will study e and ln properly in Calculus and Vectors (MCV4U).

Just for fun

Before electronic calculators, engineers carried slide rules: two rulers marked with logarithmic scales. Sliding one along the other added logarithms, which multiplied numbers. Many of the engineers of the 1960s space programs did their calculations this way, using nothing more than logarithms on a stick.

Hands-on activity

Build a slide rule

You need: two strips of paper or light cardboard about 30 cm long, a ruler, a pencil and a calculator.

On each strip, make a mark near the left end and label it 1. This is your starting point.
For each whole number n from 2 to 10, find log n on the calculator, multiply by 25, and measure that many centimetres from the 1-mark. Mark and label n. (For example, log 2 ≈ 0.301, so 2 goes at 7.5 cm; 10 goes at 25 cm.) Use exactly the same marks on both strips.
To multiply 2 × 3: slide the top strip so its 1-mark sits above the 2 on the bottom strip. Find 3 on the top strip and read the number underneath it on the bottom strip.
Try 2 × 4, 3 × 3 and 2 × 5 the same way.
Explain in one sentence why it works.

What to look for: you should read about 6, 8, 9 and 10. It works because the distance to each mark is proportional to its logarithm, and placing two lengths end to end adds them: log 2 + log 3 = log 6. That is the product law, which you will prove in lesson 4.

Your turn

Evaluate log 750 to four decimal places. Between which two whole numbers should it lie?

Evaluate log 0.06 to four decimal places.

Between which two whole numbers is log₃ 50? Then evaluate it to three decimal places.

Evaluate log₂ 1000 to three decimal places.

Evaluate log₆ 200 to three decimal places.

Evaluate log₉ 2 to three decimal places and explain why it is less than 1/2.

Evaluate log_(0.5) 10 to three decimal places. Why is it negative?

More practice: warm-up

What base does log x have when no base is written?

Evaluate log 100 000 without a calculator.

Evaluate log 2 to four decimal places.

Evaluate log 30 to four decimal places.

Write log₄ 50 using the change of base formula. Do not evaluate.

More practice: on your own

Evaluate log₄ 50 to three decimal places and check your answer.

Evaluate log₈ 3 to three decimal places.

Evaluate log₃ 0.2 to three decimal places.

Given log 2 ≈ 0.3010, estimate log 20, log 200 and log 0.2 without a calculator.

A bacteria culture triples every hour. How many hours until it is 1000 times its starting size? (Solve 3ᵗ = 1000.)

Challenge

Show that log₂ 3 × log₃ 2 = 1, and evaluate both factors.

Use the change of base formula to show that log₈ x = (1/3) log₂ x.

How many digits does 2¹⁰⁰ have? (Hint: a whole number N with d digits satisfies 10^(d−1) ≤ N < 10ᵈ.)

Check your understanding

Quiz

What does log 7 mean when no base is written?

Between which two whole numbers does log 3800 lie?

Which expression equals log₅ 12?

To three decimal places, log₂ 7 is about

A student finds log₄ 30 ≈ 0.408. Which check shows the answer is wrong?

Which value is negative?

For parents and teachers

Big idea (A1.2): a calculator's LOG key gives base-10 logarithms, and the change of base formula log_b x = log x ÷ log b gives a logarithm in any base. Estimating between whole-number powers first is the best error check.

How to help: Ask your teen to estimate before calculating ("log₃ 50 is between 3 and 4 because 27 < 50 < 81"), then to check by raising the base to the answer. If the check fails, the keystrokes were wrong.

Common difficulties:

Dividing the wrong way round (log b ÷ log x instead of log x ÷ log b). The argument goes on top.
Typing log 20 ÷ 2 instead of log(20) ÷ log(2). Brackets matter on calculators.
Rounding log values too early. Keep all the calculator's digits until the last step.
Thinking the ln key is "the same as" log. Both work in the change of base formula, but alone they give different numbers.