C2 – Exponentials and Logarithms

Key Facts:

The functions

y = a^x or f(x) = a^x

where a is a constant, are exponential functions

a^x = n means that log_an = x

where a is called the base of the logarithm

log_a1 = 0

Any number to the power 0 is 1, x⁰ = 1 or a⁰ = 1

log_aa = 1

Any number to the power 1 is just the number itself, 10¹ = 10,

6¹ = 6 or a¹ = a

Multiplication law

log_a xy = lag_a x + log_a y

Division law

log_a (x/y) = log_a x – log_a y

Power law

log_a (x)^k = k log_a x

From the power law

log_a (1/x) = -log_a x

Equations of the form a^x = b can be solved by taking logarithms (to base 10) of each side

Change of base rule

log_ax = (log_bx)/(log_ba)

From the change of base rule

log_ab = 1/log_ba

For the OCR, you need to remember all of the above key points. For Edexcel, only the change of base rule is given in the formulae booklet, everything else you need to remember.