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=(\log 1000)/(\log 8)`
`\left[\because \log_ab=(\log_x a)/(\log_x b)\right]`

`LHS=(\log (10)^3)/(\log 2^3)`

`LHS=(3\log 10)/(3\log 2)`
`[\because \log x^n=n\log x]`

`LHS=(\log 10)/(\log 2)`

`LHS=\log_210`
`\left[\because \log_ab=(\log_x a)/(\log_x b)\right]`

`LHS=RHS`

Hence proved.