A-level · Exponentials & logarithms

Solving exponential and logarithmic equationsExam-style questions

Taking logs to solve a^x = b, equations with e^x and ln, hidden quadratics in e^x or a^x, and log equations solved with the laws — always checking which roots are valid.

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14 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
Solve the equation

$3^{2x+1} = 5^{x+2}$ ,

giving your answer in the form $x = \dfrac{\ln a}{\ln b}$ , where $a$ and $b$ are rationals, and then to 3 significant figures. (5 marks)
Level 3 · 5 marksTry it
2
Solve the equation $\ln(2x+3) = 2\ln x$ . (4 marks)
Level 3 · 4 marksTry it
3
Solve the equation

$2\log_3 x - \log_3 (x + 6) = 1$ ,

explaining why one root of your quadratic must be rejected. (5 marks)
Level 3 · 5 marksTry it
4
Solve the equation

$3^{2x + 1} - 10\left(3^{x}\right) + 3 = 0$

giving your answers in exact form. (4 marks)
Level 3 · 4 marksTry it
5
(a) Show that the equation

$2\log_3(x + 1) - \log_3(x + 5) = 2$

can be written as

$x^2 - 7x - 44 = 0$

(3 marks)
(b) (i) Solve the equation $x^2 - 7x - 44 = 0$

(ii) Hence state the solution of $2\log_3(x + 1) - \log_3(x + 5) = 2$ , giving a reason why the other root of the quadratic must be rejected.

(2 marks)
Level 3 · 5 marksTry it

4 more questions on solving exponential and logarithmic equations

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