Why is this exponential function x = ln6/ln(24/5) and not x = ln6/ln(34/5)

When you are solving for this equation, you get x = ln6 / (ln(4/5) + ln(6))
We have to solve for a common denominator, so we would multiply ln6 by 5.
ln(4/5) + ln(30/5)
Now, we add
ln (34/5)

Why is the answer ln(24/5)??

The first part is true, but the second isn't. ln(4/5) + ln(30/5) is NOT ln(4/5 + 30/5), so the common denominator didn't help you.

The sum of logarithms is the logarithm of the PRODUCT, not the sum. ln a + ln b = ln (ab). That's just the log-of-a-product identity, turned around. So:

ln(4/5) + ln(30/5) = ln (4/5 * 30/5) = ln(120/25)

That's ln(24/5) after simplifying. It was easier from the start to skip the common denominator part and write:

ln(4/5) + ln 6 = ln(4/5 * 6) = ln(24/5)

The 4/5 is all inside the logarithm, there is no separate denominator.
ln(x) + ln(y) = ln(xy)

So
ln(4/5) + ln(6) = ln((4/5)*6) = ln(24/5)