1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ What have you tried? Do you know what the laws of logarithms say about $\log 4^{100}$? $\endgroup$ May 24 '20 at 13:44
1
$\begingroup$

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 $1$.

$\endgroup$
3
  • $\begingroup$ How did you get that inequality form the equation? Also, how do you know that it is 61 digits and that the first digit is 1? $\endgroup$
    – mikejacob
    May 24 '20 at 13:58
  • 1
    $\begingroup$ 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 $1$ $\endgroup$ May 24 '20 at 14:02
  • $\begingroup$ ok, i understand now. Thank you so much for your help. $\endgroup$
    – mikejacob
    May 24 '20 at 14:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.