Question

1/2log(4x+5)=logx

Answer 1

you have idea of all properties of log function,right?

Answer 2

\[ \frac{1}{2} \log(4x+5) = \log x \\ \, \\
\log(4x+5)^{\frac{1}{2}}=\log x\\
\, \
(4x+5)^{1/2}=x\\
4x+5=x^2\\
x^2-4x-5=0\\
(x-5)(x+1)=0
\\ \implies x=5 \text{ or } x=-1 \]

However, you ignore x=-1, because looking at the original equation, you notice that the LHS gives \(\frac{1}{2}\log(4(-1)+5) = \log(1) \), which doesn't equal the RHS, \(\log(-1)\) (also, you shouldn't have a negative argument for your log, at least if you are just looking at real numbers).