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<p>Without using a calculator, evaluate $\frac{\log_5 9 + 2\log_5 6 - 4\log_5 3}{\log_{25} 4}$.<span class="inline-marks">[4]</span></p>

Without using a calculator, evaluate $\frac{\log_5 9 + 2\log_5 6 - 4\log_5 3}{\log_{25} 4}$.

Laws of Logarithms

2024 CCHY P1 Q7

Exponential and Logarithmic Functions

<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 $\lg a-\lg b=\lg(a+b)$.

Solving Logarithm Equations

Domain of a Function

2025 AHS PRELIMS P1 Q9

<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 

<p>Given that $p = \log_{2} 5$ and $q = \log_{2} 3$, express $\log_{4} \left( \frac{75}{64} \right)$ in terms of $p$ and $q$.<span class="inline-marks">[5]</span></p>

Given that $p = \log_{2} 5$ and $q = \log_{2} 3$, express $\log_{4} \left( \frac{75}{64} \right)$ in terms of $p$ and $q$.

2023 ACSBR AM S3 EOY Q3

<p>Evaluate each of the following without using the calculator.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\log_{9}81$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\log_{6}\sqrt{27}+\log_{6}\sqrt{144}-\log_{6}\sqrt{18}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\log_{3}\sqrt{54}+\log_{3}\sqrt{63}-\log_{3}\sqrt{14}$</span></div></li></ol>

Evaluate each of the following without using the calculator.

$\log_{6}\sqrt{27}+\log_{6}\sqrt{144}-\log_{6}\sqrt{18}$

$\log_{3}\sqrt{54}+\log_{3}\sqrt{63}-\log_{3}\sqrt{14}$

<p>Evaluate each of the following without using a calculator.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\log_864$</div></li><li><div class="list-item-paragraph">$\log_2\sqrt4 +\log_2\sqrt3 -\log_2\sqrt6$</div></li></ol>

Evaluate each of the following without using a calculator.

<p>Simplify each of the following if possible.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\log_372 -\log_324$</div></li><li><div class="list-item-paragraph">$\log_a4m -\log_a2m +\log_a3$</div></li></ol>

Simplify each of the following if possible.

<p>Simplify $\log_525 - \log_55$ if possible.</p>

Simplify $\log_525 - \log_55$ if possible.

Quotient Law

<p>Simplify each of the following if possible. <br></p>
<ol class="abc-list"><li data-list-item-id="item-1768116347873-em9prvv1d" data-list-item-path="item-1768116347873-em9prvv1d"><div class="list-item-paragraph">$\lg2+\lg5$</div></li><li data-list-item-id="item-1768116348673-c8byscamy" data-list-item-path="item-1768116348673-c8byscamy"><div class="list-item-paragraph">$\log_3 36+\log_3\frac{1}{12}$</div></li></ol>

Simplify each of the following if possible. 


<p>Simplify  $\log_62 +\log_63$ if possible </p>

Simplify  $\log_62 +\log_63$ if possible 

<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>Given that $p={{\log }_{4}}3$ and $q={{\log }_{4}}5$, express ${{\log }_{3}}20$ in terms of $p$ and $q$.</p>

Given that $p={{\log }_{4}}3$ and $q={{\log }_{4}}5$, express ${{\log }_{3}}20$ in terms of $p$ and $q$.

<p>Given that ${{\log }_{3}}x=p$, express ${{\log }_{9}}(27\sqrt{x})$ in terms of $p$.</p>

Given that ${{\log }_{3}}x=p$, express ${{\log }_{9}}(27\sqrt{x})$ in terms of $p$.

<p>Given that $a={{2}^{m}}$, express ${{\log }_{\frac{1}{2}}}(4{{a}^{3}})$ in terms of $m$.</p>

Given that $a={{2}^{m}}$, express ${{\log }_{\frac{1}{2}}}(4{{a}^{3}})$ in terms of $m$.

<p>Given that ${{\log }_{m}}4=x$ and ${{\log }_{m}}3=y$, express ${{\log }_{\frac{1}{m}}}3$ in terms of $x$ and/or $y$.</p>

Given that ${{\log }_{m}}4=x$ and ${{\log }_{m}}3=y$, express ${{\log }_{\frac{1}{m}}}3$ in terms of $x$ and/or $y$.

<p>Given that $m={{\log }_{n}}16$, find ${{\log }_{2}}n$ in terms of $m$.</p>

Given that $m={{\log }_{n}}16$, find ${{\log }_{2}}n$ in terms of $m$.

<ol class="abc-list"><li data-list-item-id="item-1760090389426-d72uavpu6" data-list-item-path="item-1760090389426-d72uavpu6"><div class="list-item-paragraph">Given that&nbsp;$p={{\log }_{a}}9$, find ${{\log }_{3}}a$ in terms of $p$.</div></li><li data-list-item-id="item-1760090390130-33zydj06x" data-list-item-path="item-1760090390130-33zydj06x"><div class="list-item-paragraph">If&nbsp;$p=\lg 14$, express ${{\log }_{14}}1\frac{2}{5}$ in terms of $p$.</div></li></ol>

Change of Base involving Substitution

<p>Evaluate $\frac{{{\log }_{5}}4\times {{\log }_{2}}10}{{{\log }_{25}}\sqrt{10}}$ without the use of a calculator.</p>

Evaluate $\frac{{{\log }_{5}}4\times {{\log }_{2}}10}{{{\log }_{25}}\sqrt{10}}$ without the use of a calculator.

<p>Evaluate ${{\log }_{3}}49\times {{\log }_{7}}9$ without using a calculator.</p>

Evaluate ${{\log }_{3}}49\times {{\log }_{7}}9$ without using a calculator.

<p>Evaluate ${{\log }_{x}}32\times {{\log }_{64}}x$.</p>

Evaluate ${{\log }_{x}}32\times {{\log }_{64}}x$.

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