Solve: log₂(3x) × log₅(8) = log₅(a) + log₅(2a) for Variable X

$\log_23x\times\log_58=\log_5a+\log_52a$

Given a>0 , express X by a

Let's solve the problem step-by-step:

We start with the equation:

We simplify the right side using the product rule for logarithms:

Next, we simplify $\log_5 8$ on the left side:

Thus, we substitute into the original equation:

Now, divide both sides by $3 \log_5 2$:

Using the change of base formula, express $\log_5 (2a^2)$ and $\log_5 2$ with base 2:

Substitute these into the equation:

This implies:

Raising 2 to both sides of the equation to remove the logarithms:

Therefore, solving for $x$:

Thus, we conclude:

Therefore, the value of $x$ in terms of $a$ is $\sqrt[3]{\frac{2a^2}{27}}$.