Basic Mathematics & Logarithm
Logarithm Applications
Grade 11
Question:
<p>(D) If \((526)^a = (0.00526)^b = 100\), find the value of \(\frac{1}{a} - \frac{1}{b}\)</p>
<p>(p) positive integer</p>
<p>(q) negative integer</p>
<p>(r) rational but not integer</p>
<p>(s) 4</p>
Step-by-Step Solution
Key Concept: Use the relationship from exponential equations to express reciprocals of exponents in logarithmic form.
<p><strong>Step 1:</strong> From $(526)^a = 100$, taking logarithm: $a \log 526 = \log 100$</p><p><strong>Step 2:</strong> Therefore, $\frac{1}{a} = \frac{\log 526}{\log 100} = \frac{\log 526}{2}$</p><p><strong>Step 3:</strong> From $(0.00526)^b = 100$, note that $0.00526 = \frac{526}{100000}$</p><p><strong>Step 4:</strong> So $b\log(0.00526) = \log 100$, giving $b(\log 526 - 5) = 2$</p><p><strong>Step 5:</strong> Therefore, $\frac{1}{b} = \frac{\log 526 - 5}{2}$</p><p><strong>Step 6:</strong> $\frac{1}{a} - \frac{1}{b} = \frac{\log 526}{2} - \frac{\log 526 - 5}{2} = \frac{5}{2}$</p><p>∴ Answer is (r) rational but not integer.</p>
Correct Answer: q