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PSLE

PSLE Foundation Mathematics

Solving Logarithm Equations

Natural logarithm equation

Exponential and Logarithmic Functions

Logarithmic equation with a domain check

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Solve the equation $15(16^x) - 4^x = 6$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Solve the equation $\lg(2y) + 2\ln(y) = 2$.<span class="inline-marks">[4]</span></div></li></ol>

Solving Exponential Equations Using Substitution

2024 GESS PRELIMS P2 Q5

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that the solution of the equation $2^{3x+4} \times 5^{2x-1} = 16^x \times 5^{3x}$ is $\lg \frac{16}{5}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Express $2\log_2 x - \log_2(x - 4) = 3$ as a quadratic equation in $x$ and explain why there are no real solutions.<span class="inline-marks">[5]</span></div></li></ol>

Solving Exponential and Power Equations Using Logarithms

2024 FMSS PRELIMS P1 Q8

<p>Solve the equation $2 \log_{3}{p} + \frac{1}{\log_{27}{3}} - 1 = \log_{3}{(8p + 1)} $.<span class="inline-marks">[5]</span></p>

Solve the equation $2 \log_{3}{p} + \frac{1}{\log_{27}{3}} - 1 = \log_{3}{(8p + 1)} $.

2024 CGSS P1 Q3

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Solve the equation $\log_5 (3y + 1) - \frac{1}{\log_y 5} + \log_2 8 = 4$.</span><span class="inline-marks">[4]</span></div></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">By expressing the equation $\ln(6\mathrm{e}^x + 1) + x = 0$ as a quadratic equation in $\mathrm{e}^x$, solve the equation $\ln(6\mathrm{e}^x + 1) + x = 0$, giving values of $x$ in logarithmic form.</span><span class="inline-marks">[4]</span></div></li></ol>

Solve the equation $\log_5(3y+1)-\dfrac{1}{\log_y 5}+\log_2 8=4$.

By expressing the equation $\ln(6e^x+1)+x=0$ as a quadratic equation in $e^x$, solve the equation $\ln(6e^x+1)+x=0$, giving values of $x$ in logarithmic form.

(A Math) 2025 CCHY PRELIMS P2 Q6

<ol class="abc-list"></ol>
<ol class="abc-list"><li data-list-item-id="item-1772957783014-v9o9bijfb"><div class="list-item-paragraph">Solve the equation $\log_{2}(x+2) - \log_{\sqrt{2}}(x-1) = 1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-i-14"><div class="list-item-paragraph">It is given that&nbsp;$\lg a - \lg b = \lg(a+b)$.</div><ol class="i-list"><li data-list-item-id="sub-item-i"><div class="list-item-content"><span class="list-item-text">Express $a$ in terms of $b$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="sub-item-ii"><div class="list-item-content"><span class="list-item-text">State the range of $b$.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

Solve the equation $\log_2(x+2)-\log_{\sqrt{2}}(x-1)=1$.

It is given that $\lg a-\lg b=\lg(a+b)$.

Domain of a Function

Laws of Logarithms

2025 AHS PRELIMS P1 Q9

<p>Solve the equation $\log_{4} x - \frac{3}{\log_{x} 8} = (\log_{2} x)^2$, leaving your answer in terms of $\sqrt{2}$.<span class="inline-marks">[5]</span></p>

Solve the equation $\log_4 x - \frac{3}{\log_x 8} = (\log_2 x)^2$, leaving your answer in terms of $\sqrt{2}$.

2025 CGSS PRELIMS P1 Q10

<p>It is given that&nbsp;$\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain clearly why $1 &lt; x &lt; 2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$ and show that it has only one solution.</span><span class="inline-marks">[4]</span></div></li></ol>

It is given that $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$.

Hence solve $\log_{3}(4 - x^{2}) - \log_{\sqrt{3}}(x - 1) = 1$ and show that it has only one solution.

IP3 Class Test (May 2025) Q8 (2023 ACSBR AM Prelim P2 Q8)

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $\log_2 (x + 3) - \log_4 (3x + 1) = 1$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that $\ln p - \ln(p + q) = \ln q$.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $p$ in terms of $q$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the range of values of $p$ and explain clearly why $0 &lt; q &lt; 1$.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Solve the equation $\log_2 (x + 3) - \log_4 (3x + 1) = 1$.

It is given that $\ln p - \ln(p + q) = \ln q$.

2023 CSS AM S3 EOY Q9 

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} &gt; 0$ for $x &gt; 1$.</span><span class="inline-marks">[4]</span></div></li></ol>

Solve the equation $\log_2 (x^2 + 4x - 1) = 2 + 2\log_4 (3x + 2)$.

Sketch the graphs of $y = \log_2 x$ and $y = e^{2x}$ on the same axes. Use your graphs to explain why $\log_2 x + e^{2x} > 0$ for $x > 1$.

Sketching Exponential Functions and Solving Equations Graphically

Sketching Logarithmic Functions and Solving Equations Graphically

2023 Ahmad Ibrahim Sec AM S3 EOY Q8

<p>Solve the equation $\log_3 81 - 2\log_2 x + \log_2 (x + 5) = 0$.</p>

Solve the equation $\log_3 81 - 2\log_2 x + \log_2 (x + 5) = 0$.

<p>Solve each of the following equations.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\lg(x+3)=2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\log_{x}(2-x)=2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\log_{x}\left(\frac{x-x^2}{2}\right)=3$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\log_{2x}(3-11x)=2$</span></div></li></ol>

$\log_{x}\left(\frac{x-x^2}{2}\right)=3$

<p>Solve each of the following equations.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\ln4x=5$</div></li><li><div class="list-item-paragraph">$\log_x16=4$</div></li><li><div class="list-item-paragraph">$\ln(3x)=6$</div></li><li><div class="list-item-paragraph">$\log_k81=2$</div></li></ol>
<p></p>

<ol class="abc-list"><li><div class="list-item-paragraph">Solve the equation ${\log_e{(x^2 + 4x - 1)} = 2 + 2\log_e{(3x + 2)}}$<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Sketch the graphs of ${y = \log_e{x}}$ and ${y = e^{-x}}$ on the same axes. Use your graphs to</div><div class="list-item-paragraph">explain why ${\log_e{x} + e^{-x} > 0}$ for ${x > 1}$.<span class="inline-marks">[4]</span></div></li></ol>

2023 AISS AM S3 EOY Q8

<p>Given that $\ln 2=a$ and $\ln 5=b$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $\ln \sqrt[4]{50{{e}^{3}}}$ in terms of $a$ and $b$.</div></li><li><div class="list-item-paragraph">Find the value of $k$ such that $\ln k=\frac{b-3a}{2}$, give your answer to $4$ significant figures.</div></li></ol>

<p>Solve ${{\log }_{6}}({{x}^{2}}+3x+3)=0$.</p>

<p>Solve the following simultaneous equations.</p>
<p style="text-align: center">${\log_{2}(x - y) - \log_{2}(y + 4) = 2}$</p>
<p style="text-align: center; margin-left: 6.5rem">${ \: 25^{x}(5^{y}) = 0.2}$</p>

Solve the following simultaneous equations.

${\log_{2}(x - y) - \log_{2}(y + 4) = 2}$

Simultaneous Exponential Equations

<p>Solve the simultaneous equations.</p>
<p style="margin-left: 17.5rem">${ \, \frac{3^{x+1}}{3^{y}} = 9}$</p>
<p style="text-align: center">${\ln(6x + 3y) = 3\ln 2 + \ln 3}$</p>

<p>Solve the simultaneous equations.</p>
<p style="text-align: center">${e^{2}e^{x} = e^{4y}}$</p>
<p style="text-align: center">${\log_{4}(x + 2) = 1 + \log_{2}y}$</p>

Math Question Bank