Found this equation on the web: $$x^{\log25} + 25^{\log x} = 10$$
The person solved by substitution and got $x = \sqrt{10}$ which satisfies the equation.
I tried different ways after following the man's substitution method. I tried this:
$$\log\left(x^{\log25}\right) + \log\left(25^{\log x}\right) = \log 10$$
Using laws of logs:
$$\begin{align}
\log25 \cdot \log x + \log x\cdot\log 25 &= \log 10\\
1.3979 \log x + \log x (1.3979) &= 1\\
2.7958 \log x &= 1\\
\log x &= 0.357\\
x &= 10^{0.3576} \approx 2.278
\end{align}$$
This is wrong, however.
Why aren't the laws of logs holding? I'm missing something.
Sorry for asking what I'm sure is a ignorant question. I like math but am hardly an expert.
Best Answer
The problem with your approach is your first step: taking the log of both sides. When you take the log of both sides, you get $$\log(x^{\log25}+25^{\log x})=\log10$$ This is valid. However, you then somehow transform this equation into what you have written in your question: $$\log(x^{\log25})+\log(25^{\log x})=\log10$$ This is now invalid, because of the fact that $\log_b(a+c)\ne\log_ba+\log_bc$. That is to say, you can't simply split a sum inside a logarithm like what you did.