Basic Mathematics & Logarithm
Logarithm properties and equations
Grade 11
Question:
<p>If <span class="math">\(a \log_b c = 3\)</span> where <span class="math">\(a, b, c \in \mathbb{Q}\)</span>, the value of <span class="math">\(abc\)</span> is</p>
<p>(a) 9</p>
<p>(b) 12</p>
<p>(c) 16</p>
<p>(d) 20</p>
Step-by-Step Solution
Key Concept: Apply the definition of logarithm and use the given constraint to find values of a, b, and c, then compute their product.
<p><strong>Solution:</strong></p><p>Given: <span class="math">$a \log_b c = 3$</span></p><p>We have: <span class="math">$a \log_b c = 3$</span></p><p><span class="math">$= 3^{1/(1 + \log_4 3)} = 3^{1/\log_{4/3} 4} = 3^{\log_{4/3} 4}$</span></p><p>From the given conditions: <span class="math">$a = 3, b = \frac{4}{3}, c = 4$</span></p><p>Therefore: <span class="math">$abc = 3 \times \frac{4}{3} \times 4 = 16$</span></p>
Correct Answer: C