How many digits will $ab^c$ have?

Question

How many digits will $ab^c$ have?

I know that the digits of $b^c$ is calculated so:

$$\lfloor c \log_{10}b \rfloor +1$$

but what about $ab^c$ ?

Answer 1

Since $\lfloor \log_{10} x\rfloor+1$ is the number of digits of $x$, where $x$ is an integer,

it follows that $\lfloor\log_{10}ab^c\rfloor+1=\lfloor\log_{10}a+c\log_{10}b\rfloor+1$.