Standard form, exponents and logarithms

Maths: Analysis and Approaches · Number and algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Standard form writes any number as a × 10^k, where a is at least 1 and less than 10 and k is an integer. To multiply, multiply the a parts and add the powers. Then adjust if a is no longer between 1 and 10.

A logarithm is a power. log_a b asks: what power of a gives b? So a^x = b means the same as x = log_a b. Here log means base 10 and ln means base e.

Because logs are powers, the log laws copy the exponent laws: multiplying numbers adds their logs. To solve an equation with x in the power, take logs of both sides and bring the power down.

Rules to remember

a^m × aⁿ = a^{m + n}; a^m ÷ aⁿ = a^{m − n}; (a^m)ⁿ = a^mn
a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^m/n = (n-th root of a)^m
a^x = b ⇔ x = log_a b (a > 0, a ≠ 1, b > 0)
log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y
log_a(x^m) = m log_a x
log_a 1 = 0; log_a a = 1; ln(e^x) = x; e^{ln x} = x
Change of base: log_a x = (log_b x)/(log_b a), for example (ln x)/(ln a)

Common mistake

Writing log(x + y) as log x + log y. The addition law works for the log of a product, log(xy), never for the log of a sum.

Worked example

Solve 5^{2x − 1} = 12, giving x exactly and to 3 significant figures.

Take ln of both sides: (2x − 1) ln 5 = ln 12.
2x − 1 = (ln 12)/(ln 5), so x = (1/2) × (1 + (ln 12)/(ln 5)).
(ln 12)/(ln 5) = 2.4849 ÷ 1.6094 = 1.5440, so x = (1 + 1.5440) ÷ 2.

Answer: x = (1/2)(1 + (ln 12)/(ln 5)) ≈ 1.27

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Topic 1: Number and Algebra

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