Solve the Logarithmic Equation: log₃(x²)log₅(27) - log₅(8) = ln(e)

To solve this problem, we'll follow these steps:

• Step 1: Convert the logarithms into another base using the change of base rule.
• Step 2: Simplify $\ln e$ since $\ln e = 1$.
• Step 3: Simplify the expression using known values.
• Step 4: Solve the equation for $x$.

Now, let's work through each step:

Step 1: Given the equation $\log_3 x^2 \log_5 27 - \log_5 8 = \ln e$, we know that $\ln e = 1$. We will first simplify the right side to get:
$\log_3 x^2 \log_5 27 - \log_5 8 = 1$

Step 2: Use the change of base formula.

Using $\log_b a = \frac{\ln a}{\ln b}$, rewrite $\log_5 27$ and $\log_5 8$:

$\log_5 27 = \frac{\ln 27}{\ln 5} \quad \text{and} \quad \log_5 8 = \frac{\ln 8}{\ln 5}$

Plug in the values:

$\log_3 x^2 \frac{\ln 27}{\ln 5} - \frac{\ln 8}{\ln 5} = 1$

Step 3: Multiply through by $\ln 5$ to eliminate the denominators:
$\log_3 x^2 \ln 27 - \ln 8 = \ln 5$

Now knowing $\ln 27 = 3\ln 3$, solve the equation:

$\log_3 x^2 = \frac{\ln 5 + \ln 8}{3 \ln 3}$

Apply the logarithm base rule:

$x^2 = 3^{\left(\frac{\ln 5 + \ln 8}{3\ln 3}\right)}$

Step 4: Simplify and solve for $x$. Recognize this exponent could become $\frac{\ln 40}{3\ln 3}$:

$x^2 = 3^{\frac{\ln 40}{3\ln 3}} = 40^{1/3}$

Finally, solve for $x$:

$x = \pm \sqrt[6]{40}$

Therefore, the solution to the problem is $x = \pm\sqrt[6]{40}$.