Exponents and logarithms

The green curve is y = 2^x. The orange one is its mirror image in the dashed line, which is y = log₂x. Slide x and watch the same pair of numbers read both ways.

x = 5

322^x

5log₂ of that

Going up the green curve and back down the orange one returns you to where you started. That is what inverse means.

The one fact to hold on to

a^x = b is the same statement as log_ab = x. Not a related fact, the same one written the other way round. Reading a log question as “what power of a gives b” solves most of them without a formula.

2⁵ = 32⇔ log₂32 = 5 10³ = 1000⇔ log₁₀1000 = 3 e¹ = 2.7183⇔ ln e = 1 ln means log to base e a⁰ = 1⇔ log_a1 = 0, for every a

You can never take the log of zero or a negative number. No power of 10 gives −5, so log(−5) does not exist. If one appears mid-question, something earlier went wrong. And log₁₀0.5 is negative, which is fine: a number under 1 needs a negative power.

The laws of exponents

multiply→ 5³ × 5⁻⁶ = 5⁻³ = 1125 divide→ 6⁴ ÷ 6³ = 6 power of a power→ (2³)⁻⁴ = 2⁻¹² bracket everything→ (2x)⁴ = 16x⁴ the 2 is raised too zero→ 7⁰ = 1

Solving an exponential equation

given 3 × 2^t = 96 divide first→ 2^t = 32 deal with the 3 before the power read it→ t = log₂32 = 5

Get the power on its own before you take any logs. Taking the log of both sides while the 3 is still there is the error that turns a one-line question into a mess.

Worked example

A stretch of the Chao Phraya has 800 water hyacinth plants. The number grows by 15% each week.

After how many whole weeks does it first pass 10,000?

800 × 1.15^t = 10000Growth by a fixed percentage is a power, so t is in the exponent and must be got out.
1.15^t = 12.5Divide by 800 FIRST. Taking logs with the 800 still attached turns one line into three.
t = log 12.5 ÷ log 1.15A log answers "1.15 to what power?", which is exactly what is being asked.
= 18.08...Not a whole number, so it passes 10,000 part way through a week.
t = 19 whole weeksCheck both sides of it: at t = 18 there are 9,900 and at t = 19 there are 11,385. Rounding 18.08 down to 18 would be wrong, because at 18 weeks it has not yet passed.

Your turn

1. Find log₁₀1000.

2. Solve 3 × 2^t = 96.

3. log₁₀50 lies between:

0 and 1 1 and 2 10 and 100 49 and 51

Where the marks go

Isolating the power before taking logs.

Writing a⁰ = 1 and negative powers as reciprocals correctly. A negative power means one over, never a negative answer.

Checking the size of a log before accepting it. log₁₀50 must be between 1 and 2, and knowing that catches a mistyped calculator entry instantly.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

Solve 1000 × 1.05ⁿ = 2000.

Logs, to get n out of the exponent. 1.05ⁿ = 2, so n = log 2 / log 1.05 ≈ 14.2 years. Recognise it as a doubling-time question.
Which method?

Find 8 to the power two thirds.

Rational exponent: cube root then square, so 4. Bottom is the root, top is the power, and neither divides.
Which method?

A sequence is 100, 50, 25, 12.5. Does it have a sum to infinity?

Yes, geometric with r = 0.5. |r| < 1, so S = 100/0.5 = 200. Check the ratio before reaching for the formula.

The same idea elsewhere

1.9, the laws of logarithms: The three laws turn these one-line questions into ones you can solve by hand.
5.3, differentiating powers: The index laws are what make the power rule for differentiation work.

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