IGCSE Additional Mathematics (0606)

Laws of logarithms

What this topic tests, the mistakes that cost marks, and one exam-style question worked through step by step.

Key ideas and common mistakes

Recognize that log base b of a asks for the exponent to which b must be raised to get a, not the result of dividing or taking the root.
Common mistake: Confusing with division or square root leads to common errors.
Recognize that the reciprocal of log base a of b is ln a (not 1/ln a), and that adding two natural logarithms means multiplying their arguments, not adding them.
Recognize that log_a b and log_b a are reciprocals (log_a b = 1 / log_b a), so when they appear with swapped base and argument, rewrite to a single variable and solve the resulting quadratic, not treating them as independent terms.
To solve a nested logarithmic equation, undo the outer logarithm first (using its base), then solve the inner equation.
Common mistake: Students often mistakenly combine the logs or use the wrong base for exponentiation in the final step.
Rewrite both sides with the same base using change of base, bring the power 9 down, solve the resulting equation in t = log base 7 of x, and keep both the positive and negative values of t.
When solving logarithmic equations with variables as bases, rewrite both sides using the same base or logarithmic identity, and check all positive solutions except x = 1, since negative or zero bases and x = 1 are invalid for logarithms.

Write down the value of log base 2 of 64.

Try it first, then show the working

Answer: 6

12 exam-style questions on Laws of logarithms, each marked to the scheme.