Question

How can I solve 5e^3t=8e^2t with a natural log?

Answer 1

Step-by-step explanation:

$5e^{3t} = 8e^{2t} \\ \\ applying \: log_e \: on \: both \: sides: \\ \\ \therefore \: log_e(5e^{3t} ) = log_e(8e^{2t} ) \\ \\ \therefore \: log_e5 +log_e e^{3t} = log_e8 + log_e \: e^{2t} \\ \\ \therefore \: log_e5 +3t \: log_e e = log_e8 + 2t \: log_e \: e \\ \\ \therefore \: log_e5 +3t \times 1 = log_e8 + 2t \times 1 \\ ...( \because \: log_e e = 1) \\ \\ \therefore \: log_e5 +3t = log_e8 + 2t \\ \\ \therefore \:3t - 2t = log_e8 - log_e5 \\ \\ \therefore \:t= log_e \frac{8}{5}$