Maths Times Infinity > Topics > Logexp

Logarithms and exponentials. A common area of difficulty for many students. Don't worry, like everything in maths, once you "get" them, they're EASY!

Exponentials

Just a shortcut for writing things.

Instead of writing 6*6*6*6*6, we can write #6^5# to mean the exact same thing. 5 because that's how many 6's there were. All the rest, including the rules and special values can be figured out from this.

For instance, consider #7^3 \times 7^5#. Can we simplify this expression? What does it mean?

So we have three 7s multiplied by another five 7s. That gives eight 7s in total. So the answer is #7^8#.

#7^3 \times 7^5 = (7 \times 7 \times 7) \times (7 \times 7 \times 7 \times 7 \times 7) = 7^8#

This rule can be written as #a^m a^n = a^{m+n}#. Makes sense?

How about #(5^2)^3#? Well that's #(5 \times 5) \times (5 \times 5) \times (5 \times 5) = 5^6#. It's just a bunch of #5#'s timesed together, the question is how many of them are there? Well, there's #2 \times 3 = 6#.

This rule is #(a^m)^n = a^{mn}#. Neat, huh?

Logarithms

Logs are basically the opposite of powers or exponentials. In the same way that subtraction is the opposite of addition and division is the opposite of multiplication.

Okay, let's say you wanted to solve this equation:

#5x = 7#

What would you do?

Easy-peasy, just divide both sides by #5# to get #x = \frac{7}{5}#. Or just think to yourself, if you have to multiply something by 5 to get 7, it must be a fifth of seven.

Okay, now what if we changed it? Like this:

#5^x = 7#

It would be good if we could just "divide" by that #5#. It's the only thing standing between us and making #x# the subject of the equation.

Well, this is what logs are for! The answer is written like this:

#x = \log_5(7)#

Just as #7/5# means "the number that 5 has to be multiplied by to get 7", so does #\log_5(7)# mean "the number that 5 needs to be powered by to get 7".

This is the DEFINITION of log. You could check this, by experimenting with powers. Try #5^{1.2}# and compare it to #7#. Etc. Well, there are efficient algorithms for finding it (and of course computers are fast at this sort of thing) and that's what the calculator does to find the result.

Terminology

Now, we call 5 the base of the logarithm.

So, I said exponentials and logarithms were kind of opposites before. They're called inverses.

You know inverse sine? Like #\sin^{-1}#? Well, that's the inverse of #\sin# obviously. So that means that, generally speaking, if #x = \sin(y)# then actually #\sin^{-1}(x) = y#. One way of thinking of this is that you apply #\sin^{-1}# to both sides and it cancels out the #\sin# on the right.

In the same way, this happens with logs and powers. So #e^x = y# is the same as #x = \ln(y)#.