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A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Equivalent Expressions with Logarithms and Exponents

A3.1 A1.4About 70 minutes
Learning goals
  • I can expand a logarithm of a product, quotient or power of variables into simpler logarithms.
  • I can condense a sum or difference of logarithms with variables into a single logarithm.
  • I can recognize equivalent forms such as log₂ (8x) = 3 + log₂ x and 8ˣ = 2^(3x).
  • I can explain when two log expressions are equivalent only on a restricted domain, such as log (x²) and 2 log x.

Words to know

equivalent expressions
Expressions that give the same value for every allowed value of the variable.
restriction
A condition on the variable, such as x > 0, needed for an expression to be defined.
expand
Rewrite one logarithm as a sum, difference or multiple of logarithms.
condense
Rewrite a sum, difference or multiple of logarithms as one logarithm.

Same value, different look

In lesson 5 you condensed numeric expressions like log 4 + log 25 into log 100 = 2. Now the arguments contain variables. The laws are the same, but you must also keep track of restrictions: every logarithm in the expression needs a positive argument.

Why bother? When you solve equations in lessons 12 and 13, the first move is almost always to condense several logs into one, or to rewrite an exponential expression in a more useful form. Recognizing equivalent expressions (curriculum expectation A3.1) is what makes that possible.

log (x³y² ÷ z)3 log x + 2 log y − log zone logarithmsum and difference of logsexpandcondensevalid when x > 0, y > 0, z > 0
Expanding and condensing are the same laws used in opposite directions. Both forms name the same number whenever x, y and z are positive.
Worked example

Example 1: Expand

Expand fully (all variables positive): (a) log (x³y²/z), (b) log₂ (8x⁵), (c) log √(10x).

  1. (a) Quotient law, then product law, then power law: log x³ + log y² − log z = 3 log x + 2 log y − log z.
  2. (b) log₂ 8 + log₂ x⁵ = 3 + 5 log₂ x.
  3. (c) √(10x) = (10x)^(1/2), so log √(10x) = (1/2) log (10x) = (1/2)(1 + log x) = 1/2 + (1/2) log x.

(a) 3 log x + 2 log y − log z (b) 3 + 5 log₂ x (c) 1/2 + (1/2) log x

Worked example

Example 2: Condense

Write as a single logarithm (all variables positive): (a) 2 log x − log y + 3 log z, (b) log₃ (x + 2) + log₃ (x − 2), (c) (1/2) log a − 2 log b + log 5.

  1. (a) Power law first: log x² − log y + log z³. Then combine: log (x²z³/y).
  2. (b) Product law: log₃ ((x + 2)(x − 2)) = log₃ (x² − 4). (This needs x > 2 so both original logs are defined.)
  3. (c) log a^(1/2) − log b² + log 5 = log (5√a / b²).

(a) log (x²z³/y) (b) log₃ (x² − 4) (c) log (5√a/b²)

Watch the plus sign inside

log₃ (x² − 4) can be factored as log₃ ((x + 2)(x − 2)) and split. But log (x² + 4) cannot be split at all: there is no law for the log of a sum.

Equivalent only on part of the domain

The power law log_b (Mᵖ) = p log_b M needs M > 0. Look at log (x²) and 2 log x:

  • log (x²) is defined for every x ≠ 0, because x² > 0 whenever x ≠ 0.
  • 2 log x is defined only for x > 0.

They agree for x > 0, but log (x²) also has a left branch: at x = −10, log ((−10)²) = log 100 = 2, while 2 log (−10) is undefined. The correct identity for all x ≠ 0 is log (x²) = 2 log |x|. This matters when you solve equations: changing the form can add or lose solutions.

xy−8−448−22(−√10, 1)(√10, 1)y = log (x²)xy−8−448−22(√10, 1)y = 2 log xno graph for x ≤ 0
Left: y = log (x²) has two branches, mirror images in the y-axis. Right: y = 2 log x has only the right-hand branch. The right branches are identical, so the two expressions are equivalent only for x > 0.

Equivalent exponential forms

The same idea works for powers. From Grade 11: 8ˣ = (2³)ˣ = 2^(3x), and 9ˣ = 3^(2x). With logarithms, any positive base can be rewritten in terms of any other:

  • 2 = 10^(log 2), so 2ˣ = 10^(x log 2) ≈ 10^(0.301x).
  • b^(log_b x) = x and log_b (bˣ) = x.
  • log_(bⁿ) x = (1/n) log_b x. For example, log₉ x = (1/2) log₃ x.
  • log_b a = 1 ÷ log_a b. For example, log₂ 10 = 1 ÷ log 2.
Worked example

Example 3: Simplify powers and logs

Simplify (x > 0): (a) log₃ (9ˣ), (b) 10^(2 log x), (c) log₄ x + log₂ x written using base 2 only, (d) 5^(log₅ x + 1).

  1. (a) 9ˣ = 3^(2x), so log₃ (9ˣ) = 2x.
  2. (b) 2 log x = log x², so 10^(log x²) = x².
  3. (c) log₄ x = log₂ x ÷ log₂ 4 = (1/2) log₂ x. So the sum is (1/2) log₂ x + log₂ x = (3/2) log₂ x.
  4. (d) 5^(log₅ x + 1) = 5^(log₅ x) × 5¹ = 5x.

(a) 2x (b) x² (c) (3/2) log₂ x (d) 5x

Worked example

Example 4: Prove an equivalence

Show that log₂ (4x) − log₂ (x/2) = 3 for every x > 0.

  1. Quotient law: log₂ (4x ÷ (x/2)) = log₂ (4x × 2/x).
  2. Simplify the argument: 8x/x = 8.
  3. log₂ 8 = 3.

The left side always equals 3, whatever positive x you choose. Check with x = 5: log₂ 20 − log₂ 2.5 ≈ 4.3219 − 1.3219 = 3 ✓

Just for fun

log (x²) and 2 log x agree on every single positive number. Mention a negative number, though, and only one of them is still talking to you.

Hands-on activity

Equivalent or not?

You need: 12 index cards, a pencil and a scientific calculator.

  1. Write one expression on each card: log (x³); 3 log x; log (x + 10); log x + 1; log (10x); log x + log 10; log₂ (x/4); log₂ x − 2; (log x)²; 2 log x; log (x²); 2 log |x|.
  2. Sort the cards into pairs you think are equivalent, plus any cards with no partner.
  3. Test each pair with x = 2 and x = 5. If both values match, also try x = −2.
  4. Record which pairs are equivalent for all allowed x, which only for x > 0, and which not at all.

What to look for: log (x³) = 3 log x (x > 0; at x = −2 both are undefined, so they still agree). log (10x) = log x + log 10 = log x + 1, and log₂ (x/4) = log₂ x − 2 (all x > 0). log (x²) = 2 log |x| for every x ≠ 0. No partner: log (x + 10) (it is not log x + 1: at x = 2, 1.079 vs 1.301) and (log x)², which is not 2 log x (at x = 5, 0.489 vs 1.398). log (x²) and 2 log x agree at 2 and 5 but not at −2.

Your turn

1

Expand log (a²b) (a, b > 0).

2

Expand log₅ (25/x³) (x > 0).

3

Condense log x + 4 log y (x, y > 0).

4

Condense 3 log₂ m − (1/2) log₂ n (m, n > 0).

5

Simplify log₂ (16ˣ).

6

Simplify 3^(2 log₃ x) for x > 0.

7

For which x are log (x⁴) and 4 log x equivalent?

More practice: warm-up

1

Expand log (xy).

2

Expand log x⁷.

3

Condense log a − log b.

4

Simplify log (10ˣ).

5

Rewrite 4ˣ as a power of 2.

More practice: on your own

1

Expand log₃ (27x²√y) (x, y > 0).

2

Condense 2 log (x + 1) − log (x² − 1) and simplify, stating the restriction.

3

Show that log₂ x × log_x 8 = 3 for x > 0, x ≠ 1.

4

Write 5ˣ as a power of 10, with the exponent to 4 decimal places.

5

Is log (x/10) equal to log x − 1? Is it equal to (log x)/10? Explain.

Challenge

1

Simplify log₄ (x²) − log₂ x for x > 0.

2

Show that a^(log b) = b^(log a) for a, b > 0.

3

Find the error: "log (x² − 9) = log (x − 3) + log (x + 3) for all x with x² > 9."

Check your understanding

Quiz

1.

Which is log (x⁴/y) expanded (x, y > 0)?

2.

Which is 2 log a + 3 log b condensed?

3.

Simplify log₂ (32x) for x > 0.

4.

Simplify log₅ (25ˣ).

5.

For which x-values are log (x²) and 2 log x equivalent?

6.

Which expression is equivalent to 10^(log x + 2) for x > 0?

For parents and teachers

Big idea (A3.1): the laws of logarithms let you rewrite expressions with variables in equivalent forms, but each form has its own domain. log (x²) and 2 log x are equal only for x > 0.

How to help: When your teen claims two expressions are equal, ask them to test one positive and one negative value with a calculator. A mismatch on negatives is the clue to a domain restriction.

Common difficulties:

  • Splitting log (x² + 4) or log (x + 10). Only products, quotients and powers can be split.
  • Treating (log x)² as 2 log x.
  • Forgetting restrictions after condensing or expanding.
  • Mixing bases: log₄ x + log₂ x cannot be combined until both have the same base.