Article

Logarithms

Thursday, 20 November 2025

maths year-1 pure-maths year-1-pure-maths chapter-14-exponentials-and-logarithms

A logarithm is the exponent to which a base must be raised to produce a given number.

Equation

a^{x} = b x = lo g_{a} b

Usage Rules

You cannot put base numbers.
$\log_{3}1 = x$ will always be $0$ .

Mathematical Rules

lo g (a) + lo g (b) = lo g (ab) lo g (a) - lo g (b) = lo g (\frac{a}{b}) lo g (a)^{n} = n lo g (a) lo g (a^{n}) \neq = (lo g a)^{n}

$\log a$ vs $log_{a}b$

$\log a$ is short for $\log_{10}a$

Weird Exponents

If there is weird exponents, log both sides. This makes it easier to work with both sides.

3^{x} = 2^{x + 1} x lo g 3 = (x + 1) lo g 2

Tutoring Questions

7^{x + 1} = 3^{3 x + 2} (x + 1) lo g 7 = (x + 2) lo g 3 \frac{( x + 1 )}{( x + 2 )} = \frac{l o g 3}{l o g 7} \frac{( x + 1 )}{( x + 2 )} = 0.56 \dots (rounded) x + 1 = 0.56 (x + 2) x + 1 = 0.56 x + 1.12 x = 0.56 x + 1.12 - 1 x - 0.56 x = 0.12 0.44 x = 0.12 x = 0.272 \dots (repeating)

By default $\log$ is $\log_{10}$ .