It’s Professor Dave, let’s talk about logarithmic bases.
You’ve probably noticed the log button on your calculator.
But you may then wonder what base is implied by this button, since every logarithm must
have a base.
If no base is written, then it is implied to be a base ten log.
This is called the common logarithm, and we use it for a lot of practical purposes.
A typical logarithmic scale utilizes base ten logs, in the sense that one unit in such
a scale represents a tenfold increase.
This is the case with the pH scale, where one unit down means ten times more acidic,
and one unit up means ten times more basic.
But of course, there are other bases that logs can utilize.
Really, a base can be any positive number except one.
It doesn’t even have to be an integer.
Actually, it doesn’t even have to be rational.
One irrational base that we will commonly use is the natural base, E. But what is E,
exactly?
If we look at the expression one plus one over N all raised to the N power, this was
originally derived to calculate interest.
If a bank were to pay one hundred percent interest annually, we would make N equal to
one, and the expression would equal two, so you’d get back twice what you put in.
What if instead you got fifty percent interest twice a year?
N would equal two, so you get half your investment as interest after six months, and then half
of that total after another six months.
1.5 times 1.5 gives us 2.25.
As N increases, if we were to calculate interest monthly, or weekly, or daily, this expression
gets bigger, but at an ever-decreasing rate.
Each increase in N produces a smaller and smaller increase in the value of the expression,
or the additional money you’d have after a year, and as N becomes incredibly large,
we get closer and closer to a particular number but never quite get there.
The limit of this expression as N approaches infinity is called E, and it is approximately
equal to 2.718.
If N were equal to infinity, which would mean that interest was being calculated continuously,
every single instant, we would get precisely E, and this is an irrational number, meaning
that these decimals go on forever with no repeating pattern.
Any time you see E written with digits, it is therefore an approximation, no matter how
many decimals we list, so we typically just use the letter E instead.
Because E is the natural base, when we use logs of base E, they are called natural logarithms,
or natural logs.
When we use these, instead of writing log base E of X, we can abbreviate this by writing
LN of X.
If you see a button on your calculator that says LN, this is the button that takes the
natural log of a number, which is the exponent that E must be raised to in order to get that number.
E is an extremely fascinating number, and we will learn more about it when we get to
calculus, but for now, we just need to know its approximate value and how to use natural
logs, which won’t really be any different from other logs.
What if we are trying to solve equations involving logs with strange bases, how do we solve these
on our calculators?
We have a button for log base ten, and natural log, which is log base E, but we don’t have
buttons for any other base.
We have to learn a trick called the change-of-base property.
This says that if we have some log of X with base B, but we want to express this as a log
with some other base, like base A, then log base B of X will be equals to log base A of
X over log base A of B. Here is an example so we can get some context.
Say we want to express log base five of fifteen as having base ten instead, so that we can
evaluate this on the calculator.
We would just do log base ten of fifteen over log base ten of five.
This is roughly 1.176 over 0.699, or 1.682.
To verify that this is correct, let’s look at the original log.
This was asking what power five would have to be raised to in order to get fifteen.
So let’s raise five to the 1.682 power, and we do get 14.99, so given the rounding
that we performed, this looks about right.
We can also convert to natural logs this way.
Log base seven of X is equal to the natural log of X over the natural log of seven.
This skill is good to practice, so let’s check comprehension.