Exponential Equations and Change of Base Formula

Question

Solve the following equation and round your answer to the nearest thousandth.

4^{x - 1} = 3^x

Answer 1

Original equation 4^x-1 = 3^x

Take log of each side log(4^x-1) = log(3^x)

Use power of log rule (x-1)log4 = xlog3

distributive property xlog4 - log4 = xlog3

add log4 to each side xlog4 = xlog3 + log4

subtract x log3 from each side xlog4 - xlog3 = log4

factor out x x(log4 - log3) = log4

quotient rule of log x(log(4/3) = log4

division property of equality x = log4/log(4/3) = 4.819 to nearest thousandth

change of base formula (FYI) x = log_4/34

Answer 2

(x-1)log 4 = xlog 3

x(log 4 - log 3) = log 4

x=(log 4)/(log4-log 3)