To estimate this number
there are a couple of different ways to this
one way is to realize that if this
is some number a
then the equivalent form of the says
five to the a
equals nine
this is the equivalent exponential form
in this format
it may be possible with a calculator
to create some
guesses I could just make a table of
values and just guess some values for a
and within a short period time I could
narrow that down
to the nearest hundredth
a more reliable way is the following
I could take the logarithm of both sides
of this equation
the choice of logarithms is fine, if I
took log base five we would just end up
with this expression but since we're
going to be using a calculator I have to
do a numerical estimate
I'm going to choose either the natural
logarithm
or the log base ten
let's take the natural log
both sides of this equation
if I use proper use of logarithms now
I can realize that this is the same as
a times
the natural log of five
equals the natural log of nine
or that a is
the natural log of nine
divided by the natural log for log of five
at this point it becomes a calculator
exercise
if we do the calculator arithmetic
we get
one point
three six five two just taking four decimals
off the calculator
since we are interested only to the
nearest one hundredth
the answer correct to the nearest one
hundredth would be one point three seven
another way to do this problem
is to have memorized
the equation that calls for switching
bases
most people don't have that memorized it
would get us to this point immediately
but taking logarithms of both sides is a
very reliable way, it's sure-fire
and it yields the
correct answer very easily.
