Professor Dave here, let’s talk about logarithms.
As we’ve been learning about different mathematical operations, we’ve come to understand that
every operation has an inverse operation.
Addition has subtraction.
Multiplication has division.
But when it comes to exponentiation, it’s not as simple as just taking a particular
root, because roots are exponents, they’re just fractional exponents.
The square root of X squared is X because of the rules of exponents, which tell us that
X to the two to the one half equals X to the two times one half, which equals X to the
one, or simply X.
In actuality, the inverse operation of raising something to an exponent involves using something
called a logarithm.
This fact can be demonstrated if we try to find the inverse function of B to the X.
If we recall how to do this, we just swap X and Y, and then try to solve for Y.
No root will allow us to do this.
Even if we took the Y root of both sides, we would still get the Y root of X, and that
won’t help at all.
That’s where logarithms come in.
To solve for Y, we have to take the log base B of both sides.
Log base B of X equals log base B of B to the Y.
To understand why this works, let’s pause here and quickly define logarithms.
Logarithms are different ways of expressing exponential relationships.
When we say log base B of X equals Y, we are saying that B to the Y equals X.
With logs, the base of the log raised to the power of what’s on the other side of the
equals sign, will equal the number that the log is operating on.
For example, log base two of eight equals three, because two to the third is eight.
Log base five of twenty-five is two, because five squared is twenty-five.
So going back to this, taking the log base B of both sides will solve this equation,
because it’s as though we are saying that B to the log base B of X equals B to the Y.
If B to this power equals B to this power, then these two phrases must be equivalent,
and therefore Y equals log base B of X.
This is why logarithms are the inverse functions of exponents, we have defined them to be so.
The only constraint on logarithmic functions is that X must be positive, because there
is no way to raise any number to some power and get a negative number, and B must be positive
but not equal to one, because one raised to any exponent can only be one.
Now that we know what logarithms are, let’s solve a few simple logarithmic equations.
How about log base two of sixteen equals X.
What this is asking is, two raised to what power equals sixteen?
The answer is four, so X is four.
What about the log base four of X equals two.
What this is asking is, four squared equals what?
The answer is sixteen, so X is sixteen.
What if we have log base X of eighty-one equals four.
This is asking, what number, raised to the fourth power, equals eighty-one?
The answer is three, because we get three, nine, twenty-seven, eighty-one.
So we should be able to solve simple logarithmic equations with a variable in any of these
three positions.
We should also be able to convert between exponential and logarithmic forms of functions.
If we have B squared equals nine, we should be able to write this as log base B of nine
equals two, because this is the same as this base B being raised to the second power to
get this number, nine.
Evaluating logs can sometimes seem a little tricky, but you just have to think critically.
If you see something like log base seven of one over forty nine, don’t get tripped up,
just think, raising seven by what power will give me one over forty nine?
This must be negative two, because seven to the negative two is the same as one over seven
squared, which gives us one over forty nine.
Let’s also quickly look at some graphs of logarithmic equations.
We know what two to the X looks like, and we learned earlier that the inverse function
of two to the X must be log base two of X.
We also know that to graph a function’s inverse, we reflect it across the line Y equals
X, so that all the points swap their coordinates, and that means that log base two of X must
look like this.
Now instead of having a domain that includes all real numbers, it is the range that includes
all real numbers, because the base can be raised to any exponent.
However, the domain includes only positive real numbers, because raising the base to
any exponent will always produce a positive number, so this function stays to the right
of the Y axis.
Note that just the way two to the X passes through the point zero, one, log base two
of X passes through the point one, zero.
As we approach this vertical asymptote from the right, the function approaches negative
infinity, and as we go towards positive infinity, the function goes towards positive infinity.
Then from here, we can apply the transformations we already know.
Adding or subtracting a number outside of the log produces a vertical shift.
Adding or subtracting a number inside the log will produce a horizontal shift.
A negative sign on the whole logarithmic term reflects the function across the X axis, but
if it is in here, on the X itself, the function is reflected across the Y axis.
And lastly, coefficients outside and inside the log produce vertical and horizontal stretches
respectively.
Let’s check comprehension.