5^2x-7(5^x)=-12 solve for x

Question

no clue what to do. an online calculator did it by substituting 5^x for u and that makes no sense to me at all. If you turn the 5^x into 'u' and are left with u^2-7u= -12 and solve from there, isnt all the substitution doing just messing up order of operations? Like you dont want to deal with the x in the exponent place so you replace it, ignore it, do something else and then just presto, make it exist again? The computer was bugging right?

Answer 1

See the video.

Answer 2

Write the equation as (5^x)² - 7(5^x) + 12 = 0

If u = 5^x, then we have u² - 7u + 12 = 0

So, (u - 4)(u - 3) = 0

u = 4 or u = 3

Since u = 5^x, we have 5^x = 4 or 5^x = 3.

Take log of both sides of each equation:

x = log4 / log5 or log3 / log5

x ≈ 0.86 or 0.68