Question

Given log2 5 = x and log3 5 = y , express log5 12 in terms of x and y ???

The answer is x+2y/xy but i dont how to get the answer

Answer 1

`(x+2y)/(xy)`

Explanation:

Recall the Defn. of `log` function : `log_b n=m iff b^m=n.`

We are given that, `log_2 5=x, and, log_3 5=y.`

Hence, by the Defn., `2^x=5, and, 3^y=5.`

Now, `2^x=5 rArr (2^x)^(1/x)=5^(1/x) rArr 2=5^(1/x).................(1).`

Similarly, `3^y=5 rArr 3=5^(1/y).....................................(2).`

Taking, `log_5 12=z," we have, "5^z=12=2^2*3.`

Therefore, `5^z={5^(1/x)}^2(5^(1/y)),.......[because, (1), and, (2)],` or,

`5^z=5^(2/x)*5^(1/y)=5^(2/x+1/y).`

`rArr z=2/x+1/y, i.e., log_5 12=(x+2y)/(xy),` is the desired Expression.