How do you derive the uncommon properties of logarithmic function?

Question

I understand the deviation of the basic log laws such as the product,quotient,base change however how do you prove the following laws?

1. $\log_ba=\frac{1}{log_ab}$
2. Secondly how do you express a negative log in exponential form for example convert $-\log_bx$ to exponential form. NOT as an equation because I know that $-\log_bx=y$ which is equivalent to $log_b(x)=-y$ which is equivalent to $b$ to the power of $-y$ which is $x$.

Answer 1

Write $x=\log_ba$ and $y=\log_ab$. Then we get $$x=\log_ba\iff b^x=a\iff \log(b^x)=\log a\iff x\log b=\log a\iff x=\frac{\log a}{\log b}.$$ Similarly, we get $$y=\frac{\log b}{\log a}.$$ Thus, $x=\frac{1}{y}$ which proves 1.

Answer 2

1) $x = log_b a$ is equivalent to $b^x = a$. Raise both sides to the $1/x$ power.

2) What do you mean exponential form? Converting $log_b x$ to exponential form would be $b^y = x$, which is an equation.