GCSE-ready- Home
- Lessons
- IGCSE Further Pure Mathematics
- Change of base and solving equations
Change of base and solving equations
🎬 The doodle video for this lesson is coming soon. Subscribe on YouTube to see it first.
Logarithms let you bring an unknown power down to ground level. Change of base lets you work with any base on your calculator.
Solving ax = b
Take logs of both sides, then use the power law.
3x = 20
x log 3 = log 20
x = log 20log 3 = 2.73 (3 s.f.)
3x = 20
x log 3 = log 20
x = log 20log 3 = 2.73 (3 s.f.)
Change of base
loga x = logb xlogb a
A useful special case: loga b = 1logb a
log4 8 = log2 8log2 4 = 32
A useful special case: loga b = 1logb a
log4 8 = log2 8log2 4 = 32
Equations with logs in
Combine into a single log, then rewrite as a power.
log2 x + log2 (x + 2) = 3
log2 (x(x + 2)) = 3, so x(x + 2) = 23 = 8
x2 + 2x − 8 = 0, so x = 2 or x = −4
Reject x = −4: you cannot take the log of a negative number.
log2 x + log2 (x + 2) = 3
log2 (x(x + 2)) = 3, so x(x + 2) = 23 = 8
x2 + 2x − 8 = 0, so x = 2 or x = −4
Reject x = −4: you cannot take the log of a negative number.
Solve 5x + 1 = 40, giving x to 3 significant figures.
- (x + 1) log 5 = log 40
- x + 1 = log 40log 5 = 2.292...
- x = 1.292...
Answer: x = 1.29
To solve ax = b, take logs: x = log blog a. Change base with loga x = logb xlogb a. Always check log equation answers make every log defined.
Check you have got it
Answer 7 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.