Basic Mathematics & Logarithm
Applications of Logarithms
Grade Class 11
Question:
<p>Let \((x_0, y_0)\) be the solution of the equations \((2x)^{\ln 2} = (3y)^{\ln 3}\) and \(3^{\ln x} = 2^{\ln y}\). Then \(x_0\) is</p>
\(\frac{1}{6}\)
\(\frac{1}{3}\)
\(\frac{1}{2}\)
\(6\)
Step-by-Step Solution
Key Concept: Take natural logs of both equations and solve the resulting linear system in ln x and ln y. Writing both equations in logs gives two linear equations in ln x and ln y. Solving them yields x0 = 1/2.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Take natural logs of both equations and solve the resulting linear system in ln x and ln y. Writing both equations in logs gives two linear equations in ln x and ln y. Solving them yields x0 = 1/2. Trap: Do not expand powers before taking logs; the logarithmic form is linear immediately. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: C