Question

Use the properties of logarithms to prove log, 1000 = log2 10.

Answer 1

Given:

Consider the equation is:

$\log_81000=\log_210$

To prove:

$\log_81000=\log_210$ by using the properties of logarithms.

Solution:

We have,

$\log_81000=\log_210$

Taking left hand side (LHS), we get

$LHS=\log_81000$

$LHS=\dfrac{\log 1000}{\log 8}$                  $\left[\because \log_ab=\dfrac{\log_x a}{\log_x b}\right]$

$LHS=\dfrac{\log (10)^3}{\log 2^3}$

$LHS=\dfrac{3\log 10}{3\log 2}$                   $[\because \log x^n=n\log x]$

$LHS=\dfrac{\log 10}{\log 2}$

$LHS=\log_210$                    $\left[\because \log_ab=\dfrac{\log_x a}{\log_x b}\right]$

$LHS=RHS$

Hence proved.