Basic Mathematics & Logarithm
Logarithmic equations
Grade 11
Question:
<p>If <span class="math">\(x = \log_{2a} \frac{1}{bcd}, y = \log_{3b} \frac{1}{acd}, z = \log_{4c} \frac{1}{abd}, w = \log_{5d} \frac{1}{abc}\)</span> and <span class="math">\(\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} + \frac{1}{w+1} = \log_{abcd} N + 1\)</span>, the value of <span class="math">\(N\)</span> is</p>
<p>(a) 40</p>
<p>(b) 80</p>
<p>(c) 120</p>
<p>(d) 160</p>
Step-by-Step Solution
Key Concept: Use change of base formula and properties of logarithms to simplify the sum of reciprocals.
<p><strong>Solution:</strong></p><p>Given the logarithmic expressions for <span class="math">$x, y, z, w$</span></p><p>We compute: <span class="math">$\frac{1}{x+1} = \log_{2a} 2ab, \frac{1}{y+1} = \log_{3b} 3bc, \frac{1}{z+1} = \log_{4c} 4cd, \frac{1}{w+1} = \log_{5d} 5da$</span></p><p>Summing these and using properties of logarithms yields <span class="math">$N = 120$</span></p>
Correct Answer: C