This lesson is about logarithms. Let's
introduce the subject of logarithms by
considering other fields of mathematics. There are three closely related, but very
different, branches of mathematics that
are all linked to the simple expression
10 squared equals 100. Everyone knows
that 10 squared equals 100 yet what if
we replace each number one at a time
with X. For instance what if we replace
the 100 with an X then we have 10
squared equals x. Well this is simply
arithmetic: 10 squared is 10 times 10
which is 100 it's an exponent this is
part of PEMDAS. Next what if we replace
the 10 with an X now we have x squared
equals 100 and this is algebra because
in order to solve for X we have to say X
is being squared so square root both
sides and you get X is equal to 10 or
negative 10, this is algebra. However what if we
replace the exponent 2 with an X? Now we have 10 to the what equals 100? And this
type of question is a logarithm, wondering what exponent you would need
on a particular base to produce a
particular answer. So to review; case 1
demonstrate simple arithmetic you can
find X by simply multiplying, hmm that's a
typo, 10 times 10. Case 2 demonstrates
algebra you can find X by square rooting
both sides of the equation. Case 3
demonstrates the field of logarithms,
finding the necessary exponent to which
a base must be raised in order to equal
a give an answer. Most people will guess that X must be 2
on order for 10 to the X to equal 100. However what if you are given the
problem 10 to the x equals 90? The answer
is not obvious though you might guess
the answer is just less than 2
approximately 1.9. Let's take a look.
Here's our definition; a logarithm is the
exponent to which a base must be raised
in order to equal a give an answer. For
example if we wanted to solve for x in
the exponential expression 10 to the x
equals 100 we would rewrite the
expression a logarithmic expression
log base 10 of 100 equals x. So here we
read it as 10 to the power of x equals
100 but here we read it as log base 10
of 100 equals X this is an exponential
expression this is a logarithmic
expression. The helpfulness of the second
expression log base 10 of 100 is that
every scientific calculator has a log
button which means log base 10. In order
to calculate log base 10 of 100 equals x
all you need to do is type into your
calculator: the log button, then the
number 100, then enter and the answer
will clearly be 2 because 10 to the
power of what equals 100? 10 to the power of 2 equals 100. Note not all calculators
accept instructions in the same way so
double check how your calculator works.
We write logarithms in a way that groups
the base with the answer so that we can
solve for the exponent. All of these
following expressions are equivalent so
here the exponential expression 10 to
the x equals 100 means the same thing as
log base 10 of 100 equals x. We want to
practice this verbage so let's keep
going. 10 to the X is a thousand means
the same as log base 10 of a thousand is
x. Again, 10 to the X is 10,000 means the
same thing as log base 10 of 10,000
equals x so in general let us define B
as our base E as our exponent and A as
our answer. In general we find that B to
the E is A is the exponential expression
which means the same thing as log base B of A is E which is the logarithmic
expression. Use your calculator to
confirm your guess about X in the
previous examples your calculator should
verify that log base 10 of 100 is 2
because 10 squared is 100. And log base
10 of 1,000
is 3 because 10 to the power of 3
equals a thousand and log base 10 of
10,000 is 4 because 10 to the 4th equals
10,000.
Here are some practice problems. Convert the following exponential
expression into logarithmic expressions.
I'll do the first one, 10 to the X is 72
means the same as log base 10 of 72 is X
they mean the same thing X takes on the
same value in both expressions this is
our exponential expression this is our
logarithmic expression. Complete the
following examples and then go to the
next page.
Here's your answer key.
Also practice verbalizing each one, 10 to the
X is 100 log base 10 of 10 is X very
important that you use the word 'base' to
refer to the little number and 'of' to
refer to the big number so for example
log base 10 of 490 oops that's a typo
is X.
