Solve the log equation: 2log x-2 = log (x+25)

[jimmypop]

Here is what I got so far, but I'm not sure where to go from here?

log^((x^2) - log (x+25))=2
log(x^2/x+25)=2
x^2/x+25=10^2

Where do I go from here???

 

[soroban]

Hello, jimmypop!

\(\displaystyle \text{Solve: }\;2\log x - 2 \:=\:\log(x+25)\)

Note that \(\displaystyle x\) must be positive.


\(\displaystyle \text{We have: }\;\log(x^2) - \log(x+25) \:=\:2 \quad\Rightarrow\quad \log\left(\frac{x^2}{x+25}\right) \:=\:\log(10^2)\)

\(\displaystyle \text{"Un-log" both sides: }\;\frac{x^2}{x + 25} \:=\:100 \quad\Rightarrow\quad x^2 \:=\:100(x + 25)\)

. . \(\displaystyle x^2 \:=\:100x + 2500 \quad\Rightarrow\quad x^2 - 100 - 2500 \:=\:0\)

\(\displaystyle \text{Quadratic Formula: }x \:=\:\frac{100 \pm \sqrt{20000}}{2} \:=\:50(1 \pm\sqrt{2})\)

\(\displaystyle \text{Therefore: }\;x \:=\:50(1 + \sqrt{2})\)