Question

How do you write the following expression as a single logarithm: log 5 - log x - log y?

Answer 1

`log(5/(xy))`

Explanation:

We have two properties in Logarithms that can be used in solving this question

• `log_m(a) + log_m(b) = log_m(a*b)`

proof :-

Let :

`log_m(a) = p => m^p = a` ------1

and

`log_m(b) = q => m^q = b` ------2

So,

From 1 and 2 ;

`a*b = m^p * m^q`

`:. a * b = m^(p+q)`

By writing `uarr` in logarithmic form

`log_m(ab) = p + q`

`=> log_m(ab) = log_m(a) + log_m(b)`

and

• `log_m(a) - log_m(b) = log_m(a/b)`

proof:-

Let :

`log_m(a) = p => m^p = a` -------1

and

`log_m(b) = q => m^q = b` ------2

So,

From 1 and 2 ;

`a/b = m^p / m^q`

`:. a / b = m^(p-q)`

By writing `uarr` in logarithmic form

`log_m(a/b) = p - q`

`=> log_m(a/b) = log_m(a) - log_m(b)`

The given question is :

`log(5) - log (x) - log(y)`

`=> log(5) - ( log(x) + log(y) )`

`=> log(5) - ( log(x*y) )`

`=> log(5/(x*y) )`

`=> log(5/(xy) )`