Basic Mathematics & Logarithm
Properties of Logarithms
Grade 11
Question:
<p>From the relation <span class="latex">r^q = p^r</span>, find the value of <span class="latex">q</span>.</p>
<p>(a) <span class="latex">r</span></p>
<p>(b) <span class="latex">p \log_p r</span></p>
<p>(c) <span class="latex">r^{\log_r p}</span></p>
<p>(d) <span class="latex">r^r</span></p>
Step-by-Step Solution
Key Concept: Use the logarithmic relationship from <span class="latex">r^q = p^r</span> and apply change of base formula to express <span class="latex">q</span>.
<p><strong>Step 1:</strong> From <span class="latex">r^q = p^r</span>, take logarithm base <span class="latex">r</span> on both sides: <span class="latex">q = r \log_r p</span></p><p><strong>Step 2:</strong> Using logarithm properties: <span class="latex">q \log r = r \log p</span></p><p><strong>Step 3:</strong> Solving: <span class="latex">q = \frac{r \log p}{\log r} = r \log_r p</span></p><p><strong>Step 4:</strong> This can also be written as: <span class="latex">q = r^{\log_r p}</span></p><p>∴ Answer is (c) <span class="latex">r^{\log_r p}</span></p>
Correct Answer: C