logarithms and indices

Sub-topic 1 2025 · National One-Eighth Free

Metric (si) prefixes as powers of ten (true/false)

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Sub-topic 1 2026 · National Preliminary Free

Rationalising the denominator of $\dfrac{5-2\sqrt{3}}{5+2\sqrt{3}}$

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Sub-topic 2 2024 · National Preliminary Premium

Solving $\log_{3}(2x-3)=3$

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Sub-topic 4 2025 · National One-Eighth Premium

Solving $(x-3)^{x-3}=1$

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Sub-topic 5 2025 · National One-Eighth Premium

Solving $\log(x+6)+\log(x-2)=\log(2x+3)$ for a positive value of $x$

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Sub-topic 6 2024 · National Preliminary Premium

Factorising the cubics $x^3+2x^2-x-2$, $x^3-3x^2-4x+12$ and $x^3+3x^2-9x-27$

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Sub-topic 7 2025 · National One-Eighth Premium

Expressing $\log\sqrt{\dfrac{a^2b^4}{c^6}}$ in terms of $\log a$, $\log b$ and $\log c$

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Sub-topic 8 2024 · National One-Eighth Premium

Verifying $4\log x+2=\log(100x^4)$ and similar identities (true/false)

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Sub-topic 9 2024 · National Preliminary Premium

Solving $\log_{2}(x^2-8x+20)=3$

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Sub-topic 10 2026 · National Preliminary Premium

Solving $\dfrac{(2^x)^x}{8^x}=16$

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Sub-topic 11 2025 · National Semifinal Premium

$\log(mn)$ when $\log m$ and $\log n$ are the roots of $3x^2-5x+2=0$

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Sub-topic 12 2024 · National Semifinal Premium

Solving $\log_b x - b\log_x b = b-1$ using the reciprocal relationship between mutual-base logs

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Sub-topic 13 2025 · National One-Eighth Premium

Solving $(2x-3)^{x+2}=1$

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Sub-topic 14 2026 · National Semifinal Premium

Reducing an exponential relationship $y=ab^x$ to linear form $v=mu+c$

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Sub-topic 15 2026 · National Quarterfinal Premium

Solving an exponential equation of the form $\dfrac{16^x-4^x}{4^x-1}=k$ using a substitution $y=4^x$

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Sub-topic 16 2026 · National Quarterfinal Premium

Making y the subject of a logarithmic inequality of the form $\log(y-k)<2\log(ax)$

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Sub-topic 17 2026 · National Final Premium

Making y the subject of a logarithmic inequality of the form $\log(y-k)<2\log(ax)$

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Sub-topic 18 2026 · National Quarterfinal Premium

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

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Sub-topic 19 2023 · National Preliminary Premium

Solving $\log_2(x+5)=\log_2(x-2)+1$ and two similar equations

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Sub-topic 20 2023 · National Preliminary Premium

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

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Sub-topic 21 2026 · National One-Eighth Premium

Solving an exponential equation of the form $x^x=a^b$

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Sub-topic 22 2024 · Regional Championship Premium

Finding $x$ from $x^x=4^{32}$, $2^{64}$, $4^{192}$ and $5^{375}$

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Sub-topic 23 2026 · National One-Eighth Premium

Solving $\dfrac{7^{x^2}}{49^x}=343$

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Sub-topic 24 2026 · National Preliminary Premium

Solving $9^{2x-2}=3^{3x+4}$, $8^{x+4}=4^{2x-3}$ and $25^x=5^{3x-5}$

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Sub-topic 25 2018 · National Preliminary Premium

Solving $\log_2x=\log_4x$

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Sub-topic 26 2023 · National Preliminary Premium

Expressing $b$ in terms of $a$ when $\log_2a=\log_4b$

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Sub-topic 27 2025 · National Quarterfinal Premium

Solving $\log_2(x-3)+\log_2(x+3)=4$

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Sub-topic 28 2025 · National One-Eighth Premium

Solving $\log(x+6)+\log(x-2)=\log(2x+3)$

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Sub-topic 29 2023 · National One-Eighth Premium

Solving $x^x=256$

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Sub-topic 30 2021 · National Preliminary Premium

Solving $3^{\log x}=\dfrac{1}{27}$

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Sub-topic 31 2024 · National One-Eighth Premium

Verifying $4\log x+2=\log(100x^4)$ and similar identities (true/false), second contest

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Sub-topic 32 2024 · National One-Eighth Premium

Solving $\log_a x - k\log_x a = c$ using the reciprocal relationship between mutual-base logs

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Sub-topic 33 2024 · National One-Eighth Premium

Finding all real roots of $4^{x^3}=256^x$

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Sub-topic 34 2025 · National Final Premium

Solving $\log_2(x-y)=1$ and $\log_4(2x+y)=2$

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Sub-topic 35 2017 · National One-Eighth Premium

Expressing $\sqrt[5]{7^6}$, $\sqrt[3]{10^4}$ and $\sqrt[4]{23^{-5}}$ as single fractional indices

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Sub-topic 36 2026 · National One-Eighth Premium

Solving $\dfrac{7^{x^2}}{49^x}=343$ (second contest)

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Sub-topic 37 2025 · National Preliminary Premium

Solving $(2x-1)^{x+1}=1$

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Sub-topic 38 2024 · National Preliminary Premium

Expressing a decimal number in scientific notation

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Sub-topic 39 2025 · National One-Eighth Premium

Solving $a^b=1$ equations via the base=1, exponent=0, and base=-1 cases

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Sub-topic 40 2003 · National Semifinal Premium

Finding $\log(x^2)$ and $\log(\sqrt{x})$ from a given value of $\log(x)$

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Sub-topic 41 2013 · National Preliminary Premium

Solving logarithmic equations using $\log a+\log b=\log(ab)$ and $\log a-\log b=\log\left(\dfrac{a}{b}\right)$

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Sub-topic 42 2013 · National Preliminary Premium

Expressing $\log75$, $\log\dfrac{9}{125}$ and $\log\dfrac{125}{81}$ in terms of $\log5=a$ and $\log3=b$

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Sub-topic 43 2014 · National Final Premium

Evaluating $\log_5(8)\times\log_2(25)$, $\log_3(16)\times\log_2(27)$ and $\log_5(64)\times\log_4(125)$

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Sub-topic 44 2016 · National Preliminary Premium

Solving exponential equations $3^{2n+2}=81$, $5^{3n-2}=625$ and $4^{2n-3}=64$

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Sub-topic 45 2017 · National Quarterfinal Premium

Combining $\log x-3\log y+2\log z-\log t$ into a single logarithm

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Sub-topic 46 2018 · National Preliminary Premium

Combining $5\log x-3\log y+4\log z$ into a single logarithm

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Sub-topic 47 2018 · National Preliminary Premium

Finding $x+y+z$ from $\log_2(\log_3(\log_4x))=0$ and two similar equations

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Sub-topic 48 2020 · National Preliminary Premium

Solving $\log(x^2-9x)=1$

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Sub-topic 49 2020 · National One-Eighth Premium

Solving $\log_2(\log_3x)=1$ and $\log_3(\log_2x)=2$

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Sub-topic 50 2020 · National One-Eighth Premium

Solving $\log_5(\log_4x)=3$

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Sub-topic 51 2021 · National Quarterfinal Premium

Evaluating $\dfrac{\log_2 36\times\log_2 125}{\log_2 25\times\log_2 216}$

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Sub-topic 52 2021 · National Quarterfinal Premium

Solving $\log_7(3x+7)=2$

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Sub-topic 53 2021 · National Preliminary Premium

Expressing $\log(a^4b^3)$ in terms of p and q given $\log a$ and $\log b$

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Sub-topic 54 2022 · National Preliminary Premium

Problem of the day: evaluating $\dfrac1x+\dfrac1y$ given $2^x=3^y=216$, via change of base

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Sub-topic 55 2023 · National Preliminary Premium

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

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Sub-topic 56 2023 · National Preliminary Premium

Solving $\log_2(x+5)=\log_2(x-2)+1$

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Sub-topic 57 2023 · National Preliminary Premium

Solving $\log_3(2x+3)=\log_3(x-3)+1$

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Sub-topic 58 2023 · National Preliminary Premium

Solving $\log_4(x-1)+1=\log_4(2x+6)$

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Sub-topic 59 2023 · National Quarterfinal Premium

Solving the system $\log_2(2x+y)=2$, $\log_2\left(\dfrac{4x}{y}\right)=1$

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Sub-topic 60 2026 · National Preliminary Premium

Finding $m$ and $n$ from $m+n=13$ and $\log_6(mn)=2$

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Logarithms and indices

Metric (si) prefixes as powers of ten (true/false)

Rationalising the denominator of $\dfrac{5-2\sqrt{3}}{5+2\sqrt{3}}$

Solving $\log_{3}(2x-3)=3$

Solving $(x-3)^{x-3}=1$

Solving $\log(x+6)+\log(x-2)=\log(2x+3)$ for a positive value of $x$

Factorising the cubics $x^3+2x^2-x-2$, $x^3-3x^2-4x+12$ and $x^3+3x^2-9x-27$

Expressing $\log\sqrt{\dfrac{a^2b^4}{c^6}}$ in terms of $\log a$, $\log b$ and $\log c$

Verifying $4\log x+2=\log(100x^4)$ and similar identities (true/false)

Solving $\log_{2}(x^2-8x+20)=3$

Solving $\dfrac{(2^x)^x}{8^x}=16$

$\log(mn)$ when $\log m$ and $\log n$ are the roots of $3x^2-5x+2=0$

Solving $\log_b x - b\log_x b = b-1$ using the reciprocal relationship between mutual-base logs

Solving $(2x-3)^{x+2}=1$

Reducing an exponential relationship $y=ab^x$ to linear form $v=mu+c$

Solving an exponential equation of the form $\dfrac{16^x-4^x}{4^x-1}=k$ using a substitution $y=4^x$

Making y the subject of a logarithmic inequality of the form $\log(y-k)<2\log(ax)$

Making y the subject of a logarithmic inequality of the form $\log(y-k)<2\log(ax)$

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

Solving $\log_2(x+5)=\log_2(x-2)+1$ and two similar equations

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

Solving an exponential equation of the form $x^x=a^b$

Finding $x$ from $x^x=4^{32}$, $2^{64}$, $4^{192}$ and $5^{375}$

Solving $\dfrac{7^{x^2}}{49^x}=343$

Solving $9^{2x-2}=3^{3x+4}$, $8^{x+4}=4^{2x-3}$ and $25^x=5^{3x-5}$

Solving $\log_2x=\log_4x$

Expressing $b$ in terms of $a$ when $\log_2a=\log_4b$

Solving $\log_2(x-3)+\log_2(x+3)=4$

Solving $\log(x+6)+\log(x-2)=\log(2x+3)$

Solving $x^x=256$

Solving $3^{\log x}=\dfrac{1}{27}$

Verifying $4\log x+2=\log(100x^4)$ and similar identities (true/false), second contest

Solving $\log_a x - k\log_x a = c$ using the reciprocal relationship between mutual-base logs

Finding all real roots of $4^{x^3}=256^x$

Solving $\log_2(x-y)=1$ and $\log_4(2x+y)=2$

Expressing $\sqrt[5]{7^6}$, $\sqrt[3]{10^4}$ and $\sqrt[4]{23^{-5}}$ as single fractional indices

Solving $\dfrac{7^{x^2}}{49^x}=343$ (second contest)

Solving $(2x-1)^{x+1}=1$

Expressing a decimal number in scientific notation

Solving $a^b=1$ equations via the base=1, exponent=0, and base=-1 cases

Finding $\log(x^2)$ and $\log(\sqrt{x})$ from a given value of $\log(x)$

Solving logarithmic equations using $\log a+\log b=\log(ab)$ and $\log a-\log b=\log\left(\dfrac{a}{b}\right)$

Expressing $\log75$, $\log\dfrac{9}{125}$ and $\log\dfrac{125}{81}$ in terms of $\log5=a$ and $\log3=b$

Evaluating $\log_5(8)\times\log_2(25)$, $\log_3(16)\times\log_2(27)$ and $\log_5(64)\times\log_4(125)$

Solving exponential equations $3^{2n+2}=81$, $5^{3n-2}=625$ and $4^{2n-3}=64$

Combining $\log x-3\log y+2\log z-\log t$ into a single logarithm

Combining $5\log x-3\log y+4\log z$ into a single logarithm

Finding $x+y+z$ from $\log_2(\log_3(\log_4x))=0$ and two similar equations

Solving $\log(x^2-9x)=1$

Solving $\log_2(\log_3x)=1$ and $\log_3(\log_2x)=2$

Solving $\log_5(\log_4x)=3$

Evaluating $\dfrac{\log_2 36\times\log_2 125}{\log_2 25\times\log_2 216}$

Solving $\log_7(3x+7)=2$

Expressing $\log(a^4b^3)$ in terms of p and q given $\log a$ and $\log b$

Problem of the day: evaluating $\dfrac1x+\dfrac1y$ given $2^x=3^y=216$, via change of base

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

Solving $\log_2(x+5)=\log_2(x-2)+1$

Solving $\log_3(2x+3)=\log_3(x-3)+1$

Solving $\log_4(x-1)+1=\log_4(2x+6)$

Solving the system $\log_2(2x+y)=2$, $\log_2\left(\dfrac{4x}{y}\right)=1$

Finding $m$ and $n$ from $m+n=13$ and $\log_6(mn)=2$