For god's sake write the multiplication dot at the center and not at the bottom, it looks like a decimal dot.

I like yours a lot, it’s pretty creative. It’s also interesting how different your answer looks. 4^x = 9^x - 6^x x*ln(4) = ln(9^x - 6^x) | x = 0 is undefined + false-logically attempt 1 … = ln(3^x * (3^x - 2^x) ) = x*ln(3) + ln(3^x - 2^x) at first I tried factoring, but it doesn’t change the form (so these last 2 steps, ignore) attempt 2 … = ln( 9^x / 6^x - 6^x / 6^x ) | x = infinity is undefined, was already undefined = ln( 9^x / 6^x - 1 ) = ln( (9/6) ^ x - 1 ) so, 4 = (3/2) ^ x - 1 5 = (3/2) ^ x ln(5) = x*ln(3/2) x = ln(3/2 - 5) = log_5( 3/2 ) My hypothesis is, because Logarithms are homomorphisms, it seems to allow us to do the algebra Before converting / After converting (both are okay). Since log rules “look” different, the math itself will also look different. transcendental algebra is always a shot in the dark 😂, I didn’t get stuck with the sqrt, fortunately- but I did switch back-and-forth between logarithmic form

"Solving this problem needs more Thinking..." I've solved this and similar Golden Ratio problems at least 100 times and can now write down the answer without even thinking. x=lnΦ/ln(3/2)=lnφ/ln(2/3), where Φ=(√5+1)/2 and φ=(√5-1)/2. Take your pick. Nice mental exercises to stave off dementia...