Logarithms

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1. 1Logarithms and their lawsA logarithm is an index: what log means, the three laws and why they work, evaluating logs without tables, awkward bases, and solving equations with logs.Done24 min
2. 2Using four-figure tablesReading logarithm and antilogarithm tables, bar notation for numbers less than 1, and using logs to multiply, divide and take roots, as some questions still require.Done15 min

Past questions (68)

From 1983–2025: 29 WAEC, 7 NECO, 32 JAMB. Theory questions first, then objective, newest first.

Theory (20)

• WAEC 2025 · Paper 2 · Q6Using mathematical tables, find: (i) $2\sin63.35^\circ$; (ii) $\log(\cos44.74^\circ)$; (iii) $k$, given that $\log k - \log(k - 2) = \log 5$.
• NECO 2024 · Paper 2 · Q6Use mathematical tables to evaluate $\dfrac{\sqrt[4]{4000} \times 75.9}{5.61 \times 4.39}$.
• NECO 2023 · Paper 2 · Q8Evaluate without using tables $5\log2 + \log40 - \log12.8$.
• NECO 2022 · Paper 2 · Q6Use logarithm tables to evaluate $\sqrt[3]{\left(\frac{565 \times 0.0536}{249.1}\right)^2}$.
• WAEC 2022 · Paper 2 · Q13A man and his son are 42 and 12 years old respectively. How many years ago was the product of their ages 304?
• WAEC 2021 · Paper 2 · Q3A vertical pole $6\text{ m}$ high casts a shadow $9\text{ m}$ long at the same time that a tree casts a shadow $30\text{ m}$ long. Find the height …
• WAEC 2020 · Paper 2 · Q9Differentiate $y = 3t^4 - 4t^3 + 2t^2 - \dfrac1t - 5$.
• WAEC 2019 · Paper 2 · Q3Without using mathematical tables or a calculator, simplify $\dfrac{\log_2 8 + \log_2 16 - 4\log_2 2}{\log_4 16}$.
• WAEC 2019 · Paper 2 · Q4If $\log_{10} a = 1.3010$ and $\log_{10} b = 1.4771$, find the value of $ab$.
• WAEC 2019 · Paper 2 · Q4If $\cos\theta = \frac{15}{17}$, find the value of $\dfrac{\tan\theta}{1 + 2\tan\theta}$.
• WAEC 2018 · Paper 2 · Q11If $\log_a(y + 2) = 1 + \log_a x$, find $x$ in terms of $y$.
• WAEC 2017 · Paper 2 · Q1If $(y - 1)\log_{10} 4 = y\log_{10} 16$, without using mathematical tables or a calculator, find the value of $y$.
• WAEC 2017 · Paper 2 · Q7If $\log_6 y + 2\log_6 x = 3$, express $y$ in terms of $x$.
• WAEC 2016 · Paper 2 · Q9Solve: $3\log_{10}2 - 2\log_{10}3 = 1 + \log_{10}\left(\frac1x\right)$.
• WAEC 2014 · Paper 2 · Q4Without using tables or a calculator, evaluate $\log_{10}\left(\frac{75}{10}\right) - 2\log_{10}\left(\frac59\right) + \log_{10}\left(\frac{100}{243}\right)$.
• WAEC 2013 · Paper 2 · Q1Simplify, without using tables or a calculator: $\dfrac{\frac34\left(3\frac38 + 1\frac58\right)}{2\frac18 - 1\frac12}$.
• WAEC 2012 · Paper 2 · Q1Without using tables or a calculator, simplify $\dfrac{0.25 \times 3.3 \times 4200}{2.1 \times 1.65 \times 2}$, leaving your answer in standard …
• WAEC 2011 · Paper 2 · Q1Simplify: $\dfrac{\frac12 \text{ of } \frac14 \div \frac13}{\frac16 - \frac34 + \frac12}$.
• WAEC 2009 · Paper 2 · Q6If $\log 5 = 0.6990$, $\log 7 = 0.8451$ and $\log 8 = 0.9031$, evaluate $\log\left(\dfrac{35 \times 49}{40 \div 56}\right)$.
• WAEC 2008 · Paper 2 · Q4Solve the equation $2\log x - \log(1 - x) = \log(2 - x)$.

Objective (48)

• WAEC 2025 · Paper 1 · Q31If $\log_{10} x = \bar{2}.3675$ and $\log_{10} y = \bar{2}.9738$, find $x + y$, correct to three significant figures.
• WAEC 2025 · Paper 1 · Q5Given that $\log_2 a = \log_8 4$, find $a$.
• NECO 2024 · Paper 1 · Q18Evaluate $\log_4 16 + \log_3 27 - \log_8 4096$.
• WAEC 2024 · Paper 1 · Q4Express in index form: $\log_a x + \log_a y = 3$.
• WAEC 2024 · Paper 1 · Q8If $\log_3(2x - 1) = 5$, find the value of $x$.
• NECO 2023 · Paper 1 · Q7If $\log_{10}3 = 0.4771$, evaluate $\log_{10}8.1$.
• NECO 2023 · Paper 1 · Q8Given that $2\log y = 8\log p + 4\log q$, express $y$ in terms of $p$ and $q$.
• WAEC 2023 · Paper 1 · Q45If $\log_a 3 = m$ and $\log_a 5 = p$, find $\log_a 75$.
• WAEC 2023 · Paper 1 · Q6Solve $\log_3 x + \log_3(x - 8) = 2$.
• NECO 2022 · Paper 1 · Q8Given that $\log 2 = 0.30103$ and $\log 3 = 0.47712$, calculate without using tables: $\log 6 - \log 0.3$.
• WAEC 2022 · Paper 1 · Q49Given that $\log_3 27 = 2x + 1$, find the value of $x$.
• WAEC 2021 · Paper 1 · Q4If $\log_{10} 2 = m$ and $\log_{10} 3 = n$, find $\log_{10} 24$ in terms of $m$ and $n$.
• WAEC 2020 · Paper 1 · Q5Express $1 + 2\log_{10} 3$ in the form $\log_{10} q$.
• WAEC 2019 · Paper 1 · Q14Simplify $\log_{10} 6 - 3\log_{10} 3 + \frac23\log_{10} 27$.
• WAEC 2019 · Paper 1 · Q24If $\log_x 2 = 0.3$, evaluate $\log_x 8$.
• WAEC 2019 · Paper 1 · Q7Evaluate $\dfrac{\log_3 9 - \log_2 8}{\log_3 9}$.
• JAMB 2017 · UTME · Q35Find, without using logarithm tables, the value of $\dfrac{\log_3 27 - \log_{\frac14} 64}{\log_{\frac13} 81}$.
• JAMB 2016 · UTME · Q1Without using tables, evaluate $\log_2 4 + \log_4 2 - \log_{25} 5$.
• JAMB 2016 · UTME · Q26Given that $\log_4(y + 1) + \log_4\left(\frac12x\right) = 1$ and $\log_2(y - 1) + \log_2 x = 2$, solve for $x$ and $y$ respectively.
• JAMB 2015 · UTME · Q28Given that $\log_a 2 = 0.693$ and $\log_a 3 = 1.097$, find $\log_a 13.5$.
• JAMB 2015 · UTME · Q35If $\log_8 10 = x$, evaluate $\log_8 5$ in terms of $x$.
• JAMB 2014 · UTME · Q6If $\log 7.5 = 0.8751$, evaluate $2\log 75 + \log 750$.
• JAMB 2014 · UTME · Q9Evaluate $\log_2 8 + \log_2 16 - \log_2 4$.
• JAMB 2013 · UTME · Q7If $\log_{10} 4 = 0.6021$, evaluate $\log_{10} 4^{\frac13}$.
• JAMB 2012 · UTME · Q7If $\log_3 x^2 = -8$, what is $x$?
• JAMB 2011 · UTME · Q7If $\log_3 18 + \log_3 3 - \log_3 x = 3$, find $x$.
• JAMB 2010 · UTME · Q8Given that $\log 2 = 0.3010$ and $\log 7 = 0.8451$, evaluate $\log 112$.
• JAMB 2004 · UME · Q8If $6\log_x 2 - 3\log_x 3 = 3\log_5 0.2$, find $x$.
• JAMB 2003 · UME · Q7Evaluate $\log_{\sqrt2} 4 + \log_{\frac12} 16 - \log_4 32$.
• JAMB 2000 · UME · Q11Evaluate $5^{-3\log_5 2} \times 2^{2\log_2 3}$.
• JAMB 1999 · UME · Q8If $\log_8 10 = x$, evaluate $\log_8 5$ in terms of $x$.
• JAMB 1997 · UME · Q4Given that $\log_a 2 = 0.693$ and $\log_a 3 = 1.097$, find $\log_a 13.5$.
• JAMB 1997 · UME · Q5Simplify $\log_2 96 - 2\log_2 6$.
• JAMB 1995 · UME · Q7Find the value of $16^{\frac32} + \log_{10} 0.0001 + \log_2 32$.
• JAMB 1994 · UME · Q20Find the value of $r$ if $\log_{10} r + \log_{10} r^2 + \log_{10} r^4 + \log_{10} r^8 + \log_{10} r^{16} + \log_{10} r^{32} = 63$.
• JAMB 1994 · UME · Q3Evaluate $\dfrac{\log_5 0.04}{\log_3 18 - \log_3 2}$.
• JAMB 1993 · UME · Q6If $2\log_3 y + \log_3 x^2 = 4$, then $y$ is
• JAMB 1993 · UME · Q7Evaluate without using tables $\log_5 62.5 - \log_5 \frac12$.
• JAMB 1991 · UME · Q9Simplify $2\log\frac25 - \log\frac{72}{125} + \log 9$.
• JAMB 1990 · UME · Q10Simplify $3\log_6 9 + \log_6 12 + \log_6 64 - \log_6 72$.
• JAMB 1990 · UME · Q27Find the sum of the first twenty terms of the arithmetic progression $\log a, \log a^2, \log a^3, \dots$
• JAMB 1989 · UME · Q11Find $p$ in terms of $q$ if $\log_3 p + 3\log_3 q = 3$.
• JAMB 1988 · UME · Q9If $\log_{10} 2 = 0.3010$ and $\log_{10} 3 = 0.4771$, evaluate $\log_{10} 4.5$ without using logarithm tables.
• JAMB 1987 · UME · Q14Simplify without using tables $\dfrac{\log_2 6 - \log_2 3}{\log_2 8 - 2\log_2 \frac12}$.
• JAMB 1985 · UME · Q10Find $x$ if $\log_9 x = 1.5$.
• JAMB 1985 · UME · Q8Without using tables, evaluate $\log_2 4 + \log_4 2 - \log_{25} 5$.
• JAMB 1984 · UME · Q10Find without using logarithm tables the value of $\dfrac{\log_3 27 - \log_{\frac14} 64}{\log_3 \frac{1}{81}}$.
• JAMB 1983 · UME · Q33Without using tables, find the numerical value of $\log_7 49 + \log_7\frac17$.