I am confused on how to write this function as an exponential function having a base 10:

Question

P(t)= 50e^(0.5t).

The answer in my textbook is:
P(t) =50(10)^(t/4.6)

Answer 1

we want to change
e^(.5t) to 10^(kt)
let
e^(.5t) = 10^(kt)
log it
log[e^(.5t)] = log [10^(kt)]
.5t(log e) = kt(log 10)
.5 log e = k
k = .21715 or 1/4.605

so e^(.5t) = 10^(.21715t) or
e^(.5t) = 10^(t/4.605) as the have

Answer 2

e^.5t = 10^x
.5 t = ln (10^x)
.5 t = x ln 10
x = .5t/2.3 =t/4.6
so
P(t) = 50 (10)^(t/4.6)