Because the function $x\mapsto {\log }_a(x)$ is the inverse of the function $x\mapsto a^x$, there are some special properties for us to pay attention to.

Very importantly, we have

and

(This will be explained in the example.)

Additionally, we have focused so far on taking logarithms of both sides of an equation. This process can be reversed, allowing us to get rid of unwanted logarithms from our equations, for example:

Being able to manipulate logarithms and exponentials is vital, so it is important to get confident with these fundamental ideas.

1. Find, without using a calculator,

   $${\log }_5\left(\frac{1}{25}\right)$$$${\log }_7(7)$$$${\log }_{\frac{1}{2}}\left(4\right)$$

2. Solve the equation

   $${\log }_3(2x-1)=2$$