Question

Solve the expression p c^{t}=v b^{t} for t assuming that all other letters are positive constants

Answer 1

Move all the exponentials to one side, and all the constants to the other:

$\dfrac{c^t}{b^t} = \dfrac{v}{p}$

You can rewrite the left hand side as

$\left(\dfrac{c}{b}\right)^t = \dfrac{v}{p}$

Consider the logarithm base $\frac{c}{b}$ of both sides:

$log_{\frac{c}{b}}\left(\dfrac{c}{b}\right)^t = log_{\frac{c}{b}}\left(\dfrac{v}{p} \right)$

Logarithm and exponential simplifies:

$t = log_{\frac{c}{b}}\left(\dfrac{v}{p} \right)$