Question

Help my sanity please

Answer 1

`\log_a(4)`

Explanation:

You want to express this as a single logarithm with a base of 'a'.

`(log_2a(4))(1+\log_a{2})`

Rules of logarithms

The relevant rules of logarithms are ...

`\log_a(b)=(\log(b))/(\log(a))\\\\\log(ab)=\log(a)+\log(b)`

Simplification

Your expression can be simplified as follows:

`(log_2a(4))(1+\log_a{2})=(\log(4))/(\log(2a))*\left(1+(\log(2))/(\log(a))\right)\\\\\\=(\log(4))/(\log(2)+\log(a))*\left((\log(a)+\log(2))/(\log(a))\right)=(\log(4)\cdot(\log(2)+\log(a)))/(\log(a)\cdot(\log(2)+\log(a)))\\\\\\=(\log(4))/(\log(a))=\boxed{\log_a(4)}`