Question

`log_(a)(15) = m \: and \: log_(a)(5) = n \\ then \: what \: is \: \\ log_(a)( √(3) )`

options =
1)

`(1)/(2)n - m`
2)

`(1)/(2) (n - m)`
3)

`n * m`
4)

`n / m`
​

Answer 1

m-n/2

Explanation:

we are here given that,

`\longrightarrow \log_(a)(15)=m` and ,

`\longrightarrow \log_(a)(5) = n`

Taking the first given equation, we have;

`\longrightarrow \log_(a)15=m \\`

`\longrightarrow \log_(a)(5* 3) = m\\`

`\longrightarrow \log_a 5 + \log_a 3 = m`

now substitute the value from second equation,

`\longrightarrow n + \log_a 3 = m \\`

`\longrightarrow \log_a(\sqrt3)^2 = m - n \\`

`\longrightarrow 2 log_a \sqrt3 = m - n \\`

`\longrightarrow \underline{\underline{ \log_a \sqrt3 = (m-n)/(2)}}`

And we are done!