A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Simplifying and Evaluating with the Laws of Logarithms

A1.4 A3.1About 70 minutes

Learning goals

I can use several laws of logarithms in one expression to write it as a single logarithm.
I can evaluate numeric logarithmic expressions exactly, without a calculator, by combining logs first.
I can approximate logarithms of new numbers from known values such as log 2 and log 3.
I can simplify expressions such as 5^(2 log₅ 3) that combine powers and logarithms.

Words to know

single logarithm
An expression written as one log, such as log₃ 9, instead of a sum or difference of logs.

expand
Use the laws to split one logarithm into a sum, difference or multiple of simpler logs.

condense
Use the laws to combine several logarithms into a single logarithm.

prime factorization
Writing a whole number as a product of primes, such as 72 = 2³ × 3².

Putting the laws to work

In lesson 4 you proved the product, quotient and power laws one at a time. Real expressions mix them. Expressions like 2 log 5 + log 4 or log₃ 6 + log₃ 12 − log₃ 8 look like they need a calculator, but each one simplifies to a whole number. The trick is to condense first: combine everything into a single logarithm, then evaluate.

The skill also runs the other way. To estimate log 72 without the LOG key, you expand it into logs you already know.

To condense an expression: first move coefficients inside with the power law, then combine sums and differences into one log, then simplify and evaluate.

Coefficients first

In 2 log 5 + log 4, the 2 must go inside before you combine: 2 log 5 = log 5² = log 25. Writing log (5 × 4) and then doubling it is a common error.

Worked example

Example 1: Condense, then evaluate

Evaluate exactly: (a) 2 log 5 + log 4, (b) log₃ 6 + log₃ 12 − log₃ 8, (c) (1/2) log₂ 36 − log₂ 3.

(a) Power law: 2 log 5 = log 25. Product law: log 25 + log 4 = log 100 = 2.
(b) Combine left to right: log₃ (6 × 12 ÷ 8) = log₃ (72 ÷ 8) = log₃ 9 = 2.
(c) Power law: (1/2) log₂ 36 = log₂ 36^(1/2) = log₂ 6. Then log₂ 6 − log₂ 3 = log₂ 2 = 1.

(a) 2 (b) 2 (c) 1

Worked example

Example 2: Several steps

Evaluate 3 log₄ 2 + log₄ 10 − log₄ 5 exactly.

Power law: 3 log₄ 2 = log₄ 8.
Combine: log₄ (8 × 10 ÷ 5) = log₄ 16.
4² = 16, so the value is 2.

Expanding to approximate

Before calculators, people used tables of logarithms for a few key numbers and built everything else with the laws. You can do the same. Factor the number into primes, then expand.

A factor tree shows 72 = 2³ × 3². Expanding with the product and power laws turns log 72 into 3 log 2 + 2 log 3. The calculator gives 1.8573; the last digit differs only because the inputs were rounded.

Worked example

Example 3: Build logs from log 2 and log 3

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, find (a) log 1.5, (b) log 5, (c) log 45, (d) log 0.75.

(a) 1.5 = 3/2, so log 1.5 = log 3 − log 2 ≈ 0.4771 − 0.3010 = 0.1761.
(b) 5 = 10/2, so log 5 = log 10 − log 2 ≈ 1 − 0.3010 = 0.6990.
(c) 45 = 3² × 5, so log 45 = 2 log 3 + log 5 ≈ 0.9542 + 0.6990 = 1.6532.
(d) 0.75 = 3/4 = 3/2², so log 0.75 = log 3 − 2 log 2 ≈ 0.4771 − 0.6020 = −0.1249.

(a) ≈ 0.1761 (b) ≈ 0.6990 (c) ≈ 1.6532 (d) ≈ −0.1249. Check each with the LOG key.

Powers and logs together

Remember the two inverse properties from lesson 1: log_b (bˣ) = x and b^(log_b x) = x (x > 0). With the power law you can now handle expressions such as 5^(2 log₅ 3): move the 2 inside the log first.

Worked example

Example 4: Mixed powers and logs

Simplify (a) 5^(2 log₅ 3), (b) 10^(log 4 + log 2), (c) log₂ 3 × log₃ 8, (d) log₈ 32.

(a) 2 log₅ 3 = log₅ 9, so 5^(2 log₅ 3) = 5^(log₅ 9) = 9.
(b) log 4 + log 2 = log 8, so 10^(log 8) = 8.
(c) Change of base: (log 3 ÷ log 2) × (log 8 ÷ log 3). The log 3 cancels, leaving log 8 ÷ log 2 = log₂ 8 = 3.
(d) Change of base with base 2: log₈ 32 = log₂ 32 ÷ log₂ 8 = 5/3.

(a) 9 (b) 8 (c) 3 (d) 5/3

Equivalent expressions

All of these are the same number: log₂ 32, 5, log 10⁵, 5 log₂ 2 and log₈ 32 768. Recognizing equivalent forms (curriculum expectation A3.1) lets you pick the form that is easiest to work with.

Just for fun

In 1624 Henry Briggs published a book of common logarithms for tens of thousands of numbers, each worked out by hand to 14 decimal places. Your calculator does one in less than a blink. Be kind to it anyway.

Hands-on activity

Make your own log table

You need: paper, a pencil and a calculator (for checking only).

Write the numbers 1 to 20 in a column. Next to 1 write 0, next to 10 write 1.
You are allowed to look up only log 2 ≈ 0.3010, log 3 ≈ 0.4771 and log 7 ≈ 0.8451.
Using only the laws of logarithms and these three values, fill in as many other rows as you can. (For example, log 4 = 2 log 2 ≈ 0.6020; log 5 = 1 − log 2.)
Which numbers from 1 to 20 can you NOT get? Why?
Check five of your entries with the LOG key.

What to look for: you can get every number from 1 to 20 whose prime factors are only 2, 3, 5 and 7 (5 comes from 10 ÷ 2). The ones you cannot get are the primes 11, 13, 17 and 19, because no product or quotient of 2, 3, 5 and 7 equals them. Your entries should agree with the calculator to about 3 decimal places.

Your turn

Write as a single logarithm and evaluate: log₂ 12 + log₂ 6 − log₂ 9.

Evaluate 2 log 2 + 2 log 5.

Evaluate log₆ 9 + 2 log₆ 2.

Evaluate 3 log₃ 6 − log₃ 8.

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, find log 12.

Simplify 3^(2 log₃ 5).

Evaluate log₄ 8 using change of base with base 2.

More practice: warm-up

Write log 3 + log 7 as a single log.

Write 2 log 6 as a single log.

Evaluate log₅ 50 − log₅ 2.

Evaluate log₃ 3⁷.

Evaluate 10^(log 9).

More practice: on your own

Evaluate log 40 + log 50 − log 2.

Evaluate (1/3) log₂ 27 + log₂ (8/3).

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, find log 0.18.

Simplify 2^(3 log₂ 5 − log₂ 25).

Evaluate log₂₇ 81 exactly.

Challenge

Evaluate log₂ 3 × log₃ 4 × log₄ 5 × log₅ 8.

Evaluate exactly: log (1/2) + log (2/3) + log (3/4) + … + log (99/100).

If log₂ x = a, express log₈ x² in terms of a.

Check your understanding

Quiz

Evaluate log₃ 18 − log₃ 2.

Which is 3 log 2 + log 5 written as a single logarithm?

Evaluate 2 log₆ 3 + log₆ 4.

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, which is closest to log 24?

Simplify 7^(2 log₇ 4).

What is the first step in condensing 2 log x + log y − log z?

For parents and teachers

Big idea (A1.4, A3.1): the laws of logarithms let you condense several logs into one (often a whole number) or expand one log into simpler pieces. Recognizing that two expressions are equivalent is a key skill for solving equations later in the unit.

How to help: Write a messy expression like log 40 + log 50 − log 2 and ask your teen to talk you through condensing it. Insist on the order: power law first, then product and quotient. Then check with a calculator.

Common difficulties:

Combining before moving coefficients inside (writing 2 log 5 + log 4 as 2 log 20).
Sign errors with several subtractions: log a − log b − log c = log (a/(bc)), not log (ac/b).
Forgetting that 0.75 = 3/4 = 3/2² when expanding.
Thinking b^(log_b x) needs a calculator; it simply equals x.