The product of all solutions of the equation $e^{5(\log_e x)^2 + 3} = x^8$, $x > 0$, is:

Step 1: Take natural logarithm on both sides

To solve an equation where the variable is in the exponent, we can take the logarithm of both sides. Since the base of the exponential is $e$, we use the natural logarithm (base $e$). This allows us to bring down the exponent using the logarithm property $\log_b(a^c) = c \log_b a$ and $\log_e e^k = k$.

Step 2: Rearrange into a quadratic equation

Let $y = \log_e x$. Substituting $y$ into the equation transforms it into a standard quadratic form $ay^2 + by + c = 0$. This makes it easier to solve for $y$.

Step 3: Solve the quadratic equation for $\log_e x$

We solve the quadratic equation $5y^2 - 8y + 3 = 0$ by factoring. The factors are $(5y - 3)$ and $(y - 1)$, which gives us two possible values for $y$: $3/5$ and $1$.

Step 4: Solve for $x$

Now we substitute back $y = \log_e x$ and solve for $x$. Using the definition of logarithm, if $\log_b a = c$, then $a = b^c$. This gives us two solutions for $x$. Both solutions are positive, satisfying the condition $x > 0$.

Step 5: Calculate the product of solutions

The problem asks for the product of all solutions. We multiply the two values of $x$ we found. When multiplying exponential terms with the same base, we add their exponents.

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