Number of Real Roots in Logarithmic Equation

Simplify the ratios using base change rule $\frac{\log_a m}{\log_a n} = \log_n m$:

• $\frac{\log_a(16)}{\log_a(32)} = \log_{32}(16) = \frac{4}{5}$.
• $\frac{\log_a(64)}{\log_a(32)} = \log_{32}(64) = \frac{6}{5}$. The equation becomes: $\frac{4}{5}x^2 - \frac{6}{5}x - x = 0 \implies \frac{4}{5}x^2 - \frac{11}{5}x = 0 \implies x\left(\frac{4}{5}x - \frac{11}{5}\right) = 0$. So $x = 0$ or $x = \frac{11}{4}$. However, for $\log_a(x)$ terms to be defined, base $a$ must satisfy $a > 0$ and $a \neq 1$. If $a = 1$, the terms $\log_a(32)$ are undefined. Thus the valid roots depend on whether $a$ is valid ($a \ne 1$). Therefore, the answer depends on the value of $a$.