Basic Mathematics & Logarithm
Properties of Logarithms
Grade 11
Question:
<p>If <span class="latex">\log_p\{\log_q(\log_r x)\} = 0</span> and <span class="latex">10^{\log_p[\log_q(\log_r x)]} = 1</span>, find the value of <span class="latex">x</span>.</p>
<p>(a) <span class="latex">qr</span></p>
<p>(b) <span class="latex">r^q</span></p>
<p>(c) <span class="latex">r^p</span></p>
<p>(d) <span class="latex">r^q</span></p>
Step-by-Step Solution
Key Concept: Use the property that if <span class="latex">\log_a b = 0</span>, then <span class="latex">b = 1</span>, and systematically unwind the nested logarithms.
<p><strong>Step 1:</strong> From <span class="latex">10^{\log_p[\log_q(\log_r x)]} = 1</span>, we get <span class="latex">\log_p\{\log_q(\log_r x)\} = 0</span></p><p><strong>Step 2:</strong> Since <span class="latex">\log_p\{\log_q(\log_r x)\} = 0</span>, we have <span class="latex">\log_q(\log_r x) = 1</span></p><p><strong>Step 3:</strong> From <span class="latex">\log_q(\log_r x) = 1</span>, we get <span class="latex">\log_r x = q</span></p><p><strong>Step 4:</strong> From <span class="latex">\log_r x = q</span>, we obtain <span class="latex">x = r^q</span></p><p>∴ Answer is (b) <span class="latex">r^q</span></p>
Correct Answer: B