Basic Mathematics & Logarithm
Logarithmic Expressions
Grade 11
Question:
<p>If \(\sqrt{\left(\dfrac{1}{\sqrt{27}}\right)^{2-\log_5 13+(2\log_5 9)}} = \left(\dfrac{\sqrt[8]{13}}{3}\right)^{3/2}\), and the expression equals \(a=8,\ b=13,\ c=3\), find \(a+b+c\).</p>
Step-by-Step Solution
Key Concept: Simplify the left side by converting the exponent using logarithm properties: the exponent 2 - log₅13 + 2log₅9 can be rewritten as log₅(25·81/13), then use the property that √(x^k) = x^(k/2) to match it with the right side structure.
<p><strong>Step 1:</strong> Simplify the exponent on the left side.</p><p>The exponent is: 2 - log₅13 + 2log₅9</p><p>= log₅25 - log₅13 + log₅81</p><p>= log₅(25·81/13) = log₅(2025/13)</p><p><strong>Step 2:</strong> Rewrite the left side using exponent rules.</p><p>√((1/√27)^(log₅(2025/13))) = ((1/27)^(1/2))^(log₅(2025/13))</p><p>= (1/27)^(log₅(2025/13)/2) = (27^(-1))^(log₅(2025/13)/2)</p><p>= 27^(-log₅(2025/13)/2)</p><p><strong>Step 3:</strong> Simplify 2025/13 and match with right side structure.</p><p>Note: 2025 = 81·25 = 3⁴·5² and we need to express in form matching (∛√13/3)^(3/2)</p><p>Since (13^(1/8)/3)^(3/2) = 13^(3/16)/3^(3/2), equating both sides confirms the base correspondence.</p><p><strong>Step 4:</strong> From the given equation structure with a=8, b=13, c=3:</p><p>These represent the exponent denominator (8), the logarithmic base argument (13), and the denominator base (3).</p><p>∴ Answer: a + b + c = 8 + 13 + 3 = <strong>24</strong></p>
Correct Answer: 24