Indices, logarithms & surds

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1. 1Indices and exponential equationsThe General Maths index laws, pushed further: equations with several bases, fractional powers with letters, sums of powers, and equations that hide a quadratic.Done18 min
2. 2Logarithms at Further Maths depthSimplifying awkward logs without tables, changing the base (and why logₓ a = 1 ÷ logₐ x), and solving log equations, including pairs of them.Done18 min
3. 3Surds at Further Maths depthRationalising a denominator with two surd terms, writing answers in the form p + q√r, working with x = a + √b, and solving equations with square roots in them.Done18 min

Past questions (47)

From 2008–2023: 43 WAEC, 4 NECO. Theory questions first, then objective, newest first.

Theory (34)

• NECO 2023 · Paper 2 · Q2Find the equation of the circle whose centre is $(4, -5)$ and which passes through $(-3, 2)$.
• WAEC 2023 · Paper 2 · Q3Express $\dfrac{7 + 2\sqrt{10}}{9\sqrt5 + 5\sqrt2}$ in the form $p\sqrt5 + q\sqrt2$.
• WAEC 2022 · Paper 2 · Q2Solve $2^{(2y + 1)} - 5(2^y) + 2 = 0$.
• WAEC 2022 · Paper 2 · Q9Evaluate $\dfrac{7}{2 - \sqrt3} - \dfrac{7}{2 + \sqrt3}$, leaving the answer in the form $p + q\sqrt n$, where $p$, $q$ and $n$ are real numbers.
• WAEC 2020 · Paper 2 · Q10If $15^y \times 5^{(2y - 2)} \times 9^{(y - 1)} = 1$, find the value of $y$.
• WAEC 2019 · Paper 2 · Q1Simplify: $\dfrac{625^{\left(\frac{3x}{4} - 1\right)} + 125^{(x - 1)}}{5^{(3x - 2)}}$.
• WAEC 2019 · Paper 2 · Q2Find the coordinates of the point which divides the line joining $(7, -5)$ and $(-2, 7)$ externally in the ratio $3 : 2$.
• WAEC 2019 · Paper 2 · Q9Solve the equations $3\log_2 x = y$ and $\log_2 4x = y + 4$ simultaneously.
• WAEC 2019 · Paper 2 · Q10The lengths of the sides of a triangle, in centimetres, are $(3 + \sqrt2)$, $2\sqrt3$ and $3\sqrt2$. Find the value of the largest angle of the …
• WAEC 2018 · Paper 2 · Q10Solve $2^{3y + 2} - 7(2^{2y + 2}) - 31(2^y) - 8 = 0$, $y \in \mathbb R$.
• WAEC 2018 · Paper 2 · Q2Given that $\log_3 x - 3\log_x 3 + 2 = 0$, find the values of $x$.
• WAEC 2018 · Paper 2 · Q1If $\dfrac{\tan\theta + \tan30^\circ}{1 - \tan\theta\tan30^\circ} + 1 = 0$, find $\tan\theta$, leaving the answer in surd form.
• WAEC 2018 · Paper 2 · Q11Given the matrix $A = \begin{pmatrix} 3 & 8 \\ 5 & -2 \end{pmatrix}$, find its inverse.
• WAEC 2018 · Paper 2 · Q4If $\sqrt{3x + 1} - \sqrt{2x - 1} = 1$, find the values of $x$.
• WAEC 2018 · Paper 2 · Q1If $x = 2 + \sqrt5$, find the value of $x - \dfrac1x$ and leave the answer in the form $m + n\sqrt5$.
• WAEC 2017 · Paper 2 · Q3Given that $\log_{10} p = a$, $\log_{10} q = b$ and $\log_{10} s = c$, express $\log_{10}\left(\dfrac{p^{\frac12}q^4}{s^2}\right)$ in terms of …
• WAEC 2017 · Paper 2 · Q2The $n$th term of an Arithmetic Progression (A.P.) is given by $U_n = (3n - 2)\log5$. Find the sum of the first $n$ terms of the Arithmetic Progression …
• WAEC 2016 · Paper 2 · Q3If $f(x + 2) = 6x^2 + 5x - 8$, find $f(5)$.
• WAEC 2016 · Paper 2 · Q9Without using mathematical tables or a calculator, evaluate $\dfrac{\frac32\log27 - 3\log5\sqrt5}{\log0.6}$.
• WAEC 2016 · Paper 2 · Q1If $8^m + 2^{3m} = \frac14$, find the value of $m$.
• WAEC 2016 · Paper 2 · Q11Without using mathematical tables or a calculator, solve $2 + \log_{10} x - \log_{10} 20 = \log_{10}(x^2 + 4)$.
• WAEC 2015 · Paper 2 · Q1A binary operation $\Delta$ is defined on the set of real numbers $\mathbb R$ by $a\,\Delta\,b = a^3 - b^3$. Without using a calculator, find …
• WAEC 2014 · Paper 2 · Q1Solve $(\log_2 m)^2 - \log_2 m^3 = 10$.
• WAEC 2014 · Paper 2 · Q10Express $\dfrac{5 + \sqrt2}{3 - \sqrt2} - \dfrac{5 - \sqrt2}{3 + \sqrt2}$ in the form $a + b\sqrt2$.
• WAEC 2014 · Paper 2 · Q1Given $f(y) = \log_2\left(\dfrac{y + 1}{y + 3}\right)$:
• WAEC 2012 · Paper 2 · Q1Without using tables or calculators, simplify $\dfrac{\log7\sqrt7 - \log5\sqrt5}{\log1.4}$.
• WAEC 2012 · Paper 2 · Q10Simplify $\dfrac{4\sqrt7 + 3\sqrt2}{10 + 2\sqrt{14}}$.
• WAEC 2011 · Paper 2 · Q4Solve $2^{(2y + 2)} - 9(2^y) = -2$.
• WAEC 2011 · Paper 2 · Q3Solve for $x$ and $y$ in the equations: $\log(x - 1) + 2\log y = 2\log3$; $\log x + \log y = \log6$.
• WAEC 2011 · Paper 2 · Q4Find the third term of the exponential sequence (GP) $(\sqrt2 - 1), (3 - 2\sqrt2), \ldots$
• WAEC 2009 · Paper 2 · Q1Solve $2x^2 + x - 6 < 0$.
• WAEC 2009 · Paper 2 · Q2Solve the simultaneous equations $\log_2 x - \log_2 y = 2$, $\log_2 (x - 2y) = 3$.
• WAEC 2009 · Paper 2 · Q1Solve $2^{2x + 2} - 9(2^x) = -2$.
• WAEC 2008 · Paper 2 · Q4Solve $2^{x + 1} - 4\left(2^{-x}\right) - 7 = 0$.

Objective (13)

• NECO 2023 · Paper 1 · Q2Rationalize $\dfrac{2}{\sqrt5 + \sqrt7}$.
• NECO 2023 · Paper 1 · Q4Simplify $\frac12\log_4 64 + \log_4 16 - \log_4 2$.
• NECO 2023 · Paper 1 · Q8Solve $(243)^{2x + 1} = \dfrac{81^{x - 2}}{3^x}$.
• WAEC 2023 · Paper 1 · Q1Express $\dfrac{3}{3 - \sqrt{6}}$ in the form $x + m\sqrt{y}$.
• WAEC 2023 · Paper 1 · Q18Simplify: $\dfrac{\log\sqrt{27} - \log\sqrt{8}}{\log 3 - \log 2}$.
• WAEC 2023 · Paper 1 · Q2If $\left(\dfrac{1}{9}\right)^{2x - 1} = \left(\dfrac{1}{81}\right)^{2 - 3x}$, find the value of $x$.
• WAEC 2022 · Paper 1 · Q19Solve: $3^{2x-2} - 28\left(3^{x-2}\right) + 3 = 0$.
• WAEC 2022 · Paper 1 · Q2Simplify: $\left(\dfrac{3\sqrt{6} + \sqrt{54}}{\sqrt{5}\left(3\sqrt{5}\right)}\right)^{-1}$.
• WAEC 2022 · Paper 1 · Q3If $\log_{10}(3x - 1) + \log_{10} 4 = \log_{10}(9x + 2)$, find the value of $x$.
• WAEC 2022 · Paper 1 · Q4Simplify: $\dfrac{9 \times 3^{n+1} - 3^{n+2}}{3^{n+1} - 3^{n}}$.
• WAEC 2020 · Paper 1 · Q12If $V = P\log_X(M + N)$, express $N$ in terms of $X$, $P$, $M$ and $V$.
• WAEC 2020 · Paper 1 · Q39If $\log_5\left(\dfrac{125x^3}{\sqrt[3]{y}}\right)$ is expressed in the form $p\log_5 x + q\log_5 y + k$, find the values of $p$, $q$ and $k$ …
• WAEC 2020 · Paper 1 · Q7Simplify: $\dfrac{\sqrt{5} + 3}{4 - \sqrt{10}}$.