Converting log form of equation into linear form

Question

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert $10^{4 log (x)}$, into $x^4$, but I'm really unsure of how to get from this first stage to the second. My experience with logarithms and exponents is limited, though I believe that $10^{4 log (x)}$ can be re-written as $10^{4} + 10^{log(x)}$, but I'm not sure that this helps my plight! Any very basic guidance would be greatly appreciated.

Answer 1

A cool rule about logarithmic functions is that $$a\cdot \log_b(x)=\log_b(x^a).$$ From this, $$10^{4\log(x)}=10^{\log(x^4)}.$$ If we assume that you have a logarithm to the base 10, then $$10^{\log_{10}(x^4)}=x^4.$$

Answer 2

Taking logarithms, $$ \log{(10^{4\log{x}})} = 4\log{x}\log{10} = \log{x}(4\log{10}) = \log{(x^{4\log{10}})}, $$ so you can exponentiate both sides to get to $$ 10^{4\log{x}} = x^{4\log{10}}. $$ I shall spare you the rant about using $\log$ to mean $\log_{10}$.