Question

If \(4 \log_2 x - 4 \log_3 x - 16 x + 68 = 0\), then \(x - 2\) equals ………...

Hint:

When solving logarithmic equations with different bases, you can either apply logarithmic identities or use numerical methods to approximate the value of \(x\).

Answer 1

Step 1: Simplify the logarithmic terms. The given equation is: \[ 4 \log_2 x - 4 \log_3 x - 16x + 68 = 0 \] Factor out the common factor of 4: \[ 4 (\log_2 x - \log_3 x) - 16x + 68 = 0 \] Step 2: Apply properties of logarithms. The equation involves logarithms with different bases. To simplify it, we use the change of base formula for logarithms. However, we can solve this problem numerically as well. By solving this equation, we find that: \[ x = 6 \] Step 3: Calculate \(x - 2\). Thus, \(x - 2 = 6 - 2 = 4\). So, the correct answer is: \[ \boxed{6} \]