An Exponential Equation With 2 Variables

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[@dorkmania]

For the first method, after getting (a/b) in terms of log, it makes things simpler to express 12 and 18 as powers of 2 and 3.
log(12•18)= log(2³•3³)
log(18²/12) = log((3⁴•2²)/(3•2²)) = log(3³)
log(2³•3³)/log(3³)
= log(6³)/log(3³)
= log(6)/log(3)
= log_3(6)
So a/b = log_3(6)
=> 3^(a/b) = 6

[@dan-florinchereches4892]

I think the most straightforward would be grouping powers
12^a*12^b=18^(2a)/18^b
12^a*(12*18)^b=4^a*81^a dividing by 12^a
(12*18)^b = 27^a
Raising to power 1/3b
3^(a/b) = (8*27)^1/3
3^(a/b)=6

[@trojanleo123]

3^(a/b) = 6
I used the brute force method, a.k.a. the "no pain no gain" method. Lol.

[@mystychief]

a=b=0 is the solution with 3^(a/b) not defined, because the first equation means 2^(2a+2b) 3^(a+b) = 3^(4a-2b) 2^(2a-b) so that a+b=4a-2b or a=b and 2a+2b=2a-b or b=0. So the only solution is a=b=0 (meaning 12^0=18^0 which is OK) and 3^(a/b) is undefined.

[@pwmiles56]

I used method 2, streamlined a little.
12^(a + b) = 18^(2a - b)
Take both sides to the power 1/b
12^(a/b + 1) = 18^(2a/b - 1)
Separate out constant factors
12.12^(a/b) = 18^(2a/b) / 18
216.12^(a/b) = 324^(a/b)
Group terms with the same exponent
216 = (324/12)^(a/b)
216 = 27^(a/b)
Take cube roots
6 = 3^(a/b)

[@ShortsOfSyber]

Nice!

[@SidneiMV]

12^(a/b + 1) = 18^(2a/b - 1)
(18²/12)^(a/b) = 12¹18¹
27^(a/b) = 216
3^(3a/b) = 6³
*3^(a/b) = 6*

[@prollysine]

a+b=log16/log12 * (2a-b) ,

[@adamrussell658]

or not 2b!

[@key_board_x]

12^(a + b) = 18^(2a - b)
(2 * 2 * 3)^(a + b) = (2 * 3 * 3)^(a + a - b)
2^(a + b) * 2^(a + b) * 3^(a + b) = 2^(a + a - b) * 3^(a + a - b) * 3^(a + a - b)
2^(a) * 2^(b) * 2^(a) * 2^(b) * 3^(a) * 3^(b) = 2^(a) * 2^(a) * 2^(- b) * 3^(a) * 3^(a) * 3^(- b) * 3^(a) * 3^(a) * 3^(- b)
2^(b) * 2^(b) * 3^(b) = 2^(- b) * 3^(a) * 3^(- b) * 3^(a) * 3^(a) * 3^(- b)
2^(b) * 2^(b) / 2^(- b) = 3^(a) * 3^(- b) * 3^(a) * 3^(a) * 3^(- b) / 3^(b)
2^(b) * 2^(b) * 2^(b) = 3^(a) * 3^(- b) * 3^(a) * 3^(a) * 3^(- b) * 3^(- b)
2^(b + b + b) = 3^(a - b + a + a - b - b)
2^(3b) = 3^(3a - 3b)
2^(3b) = 3^[3.(a - b)]
[2^(b)]^(3) = [3^(a - b)]^(3)
2^(b) = 3^(a - b)
2^(b) = 3^(a) * 3^(- b)
3^(a) = 2^(b) / 3^(- b)
3^(a) = 2^(b) * 3^(b)
3^(a) = 6^(b)
[3^(a)]^(1/b) = [6^(b)]^(1/b)
3^(a/b) = 6^(b/b)
3^(a/b) = 6