precalculus question ? anyone help ?

Question

i had a question on my math test today and i couldnt figure it out . i was wondering if anyone could show me how it is solved ?

this is it:

3^(x-1) = 5^x

note* this was a multiple choice question and all the answers had a natural log (ln) in them

any help at all would be greatly appreciated

thanks :}

Answer 1

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We are asked to solve: 3^(x - 1) = 5^x. First we have to take the natural logarithm of both sides of the equations and apply the properties of logarithms. Recall

ln(xy) = ln(x) + ln(y)

ln(x/y) = ln(x) - ln(y)

ln(x^y) = y ln(x)

ln(3^(x - 1)) = ln(5^x)

(x - 1) ln(3) = x ln (5)

x ln(3) - ln(3) = x ln (5)

x ln(3) - x ln(5) = ln(3)

x (ln(3) - ln(5)) = ln(3)

x = ln(3) / (ln(3) - ln(5)) = -2.15066

Hope all this helps.