Which of the following shows the true solution to the logarithmic equation below logx logx 5 log6x 12?

To solve the logarithmic equation log_x(5) = log_6x(12), we will start by rewriting the equation using the change of base formula. The change of base formula states that log_a(b) = log_c(b) / log_c(a) for any base c.

First, let’s express both logarithms in the same base; we can use the natural logarithm (ln) for simplicity:

log_x(5) = ln(5) / ln(x)

log_6x(12) = ln(12) / ln(6x)

Now, substituting these into our equation gives us:

ln(5) / ln(x) = ln(12) / ln(6x)

Next, we can cross-multiply:

ln(5) * ln(6x) = ln(12) * ln(x)

Now, we can expand ln(6x):

ln(6) + ln(x)

Substituting this back into the equation gives:

ln(5) * (ln(6) + ln(x)) = ln(12) * ln(x)

This simplifies to:

ln(5) * ln(6) + ln(5) * ln(x) = ln(12) * ln(x)

Rearranging the terms provides us with:

ln(5) * ln(6) = ln(x) * (ln(12) – ln(5))

From here, we can solve for ln(x):

ln(x) = ln(5) * ln(6) / (ln(12) – ln(5))

Finally, taking the exponential of both sides gives us:

x = e^(ln(5) * ln(6) / (ln(12) – ln(5)))

This is the solution to the logarithmic equation. Therefore, the true solution is given by this expression for x.