# How do you find the number of digits for a solution for logarithmic equation and inequalities?

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.

• What have you tried? Do you know what the laws of logarithms say about $\log 4^{100}$? May 24 '20 at 13:44

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 $$1$$.
• because $1<10^{0.2...}<2$. May help you to consider if $1\times10^2<x<2\times10^2$ then $x$ has $3$ digits and its first digit is $1$ May 24 '20 at 14:02