Three Related Ideas

We have just seen how it is possible to write one number as a power of another number. This process can be related to the following two processes:

Solving Exponential Equations

is an exponential equation. Solving this equation for x is equivalent to writing 81 as a power of 3. We call it an exponential equation because the unknown is in the index. Another name for index is exponent.

This is quite different from where the unknown is in the base. With the unknown in the base we are able to solve the equation using only the third law of indices. Powers undo powers. For we only need to take a cubed root (and cubed roots are powers remember). However powers will not undo exponential equations. For this we need something new.

Undoing Exponentials

The base 3 exponential operation (that is ) turns 4 into or 81. Writing 81 as a power of 3 is equivalent to undoing this operation, that is turning 81 back into 4. This is called an inverse operation. It looks like this.

Subtraction is the inverse of addition, multiplication is the inverse of division, square root is the inverse of squaring and so on. When solving equations we use inverse operations all the time. For example, to solve the equation we subtract 5 to undo the add 5 in the equation. Similarly to solve the equation we need to undo the exponential operation .

On the previous page we solved many exponential equations. In doing so we undid many exponential operations. This included the final equation for which you found the solution (rounded to 4 decimal places).

Logarithms at last!

Because this process of undoing exponentials is so important we give it a name. We call it logarithms.

Since we say that 4 is the base 3 logarithm of 81 and write . Notice that both exponentials and logarithms have bases. To undo a base three exponential you need a base three logarithm. Notice also that the base is written as a subscript and this subscript should be read as part of the name.

How to read the logarithm notation!

Since we say that 3.5609 is the base three logarithm of 50 and write .

In terms of equations it looks like this.

The equation has the answer .

And the equation has the answer .

Logarithms on your calculator

Your calculator has two logarithm buttons. They are and . The first one is base ten and the second one uses base e . If you have never heard of e don't worry. It is used especially with calculus and you will learn it soon. There are no logarithm buttons for other bases. This is because all logarithms are much the same so you can evaluate any logarithm with any other logarithm. I will tell you the formula now and later we will see why this works.

Notice that I have not put any base on the right. This is because you can use either of the two logarithm buttons (the same one both times of course). Try it now for . That was a bit easier than the trial and error method wasn't it?