In this question using the laws of logarithms to solve the equation and inequality you're not supposed to use a calculator:
You are given that log₁₀ 4 = 0.60206 correct to 5 decimal places and that 10 (to the power of 0.206) < 2.
a) Find the number of digits in the number 4¹⁰⁰.
b) Find the first digit in the number 4¹⁰⁰.
The logarithm base $10$ of $4^{100}$ is $60.2...$,
so $4^{100}=10^{0.2...}\times10^{60}$,
i.e., $1\times10^{60}<4^{100}<2\times10^{60}$,
so $4^{100}$ has $61$ digits, and the first digit is $1$.