解方程 $3\log_5 y - \log_y 5 = 2$ 。

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完整解法

1. Apply the change of base formula to express all logarithms in the same base.

$\log_y 5 = \frac{1}{\log_5 y}$

The equation becomes:
$3\log_5 y - \frac{1}{\log_5 y} = 2$

2. Use substitution to form a quadratic equation.

Let $u = \log_5 y$ .
$3u - \frac{1}{u} = 2$
$3u^2 - 1 = 2u$
$3u^2 - 2u - 1 = 0$

3. Solve the quadratic equation for $u$.

$(3u + 1)(u - 1) = 0$
$u = -\frac{1}{3}$ or $u = 1$

4. Solve for $y$ and check validity.

When $u = 1$ :
$\log_5 y = 1 \implies y = 5^1 = 5$

When $u = -\frac{1}{3}$ :
$\log_5 y = -\frac{1}{3} \implies y = 5^{-\frac{1}{3}}$ (or $\frac{1}{\sqrt[3]{5}}$ )

Both values are positive and $y \neq 1$ , so both are valid.

答案

$y = 5$ or $y = 5^{-\frac{1}{3}}$

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