Transcript 4-3: Common and Natural Logarithms

4-3: Common and
Natural Logarithms
English Casbarro
Unit 4: Exponents/Logarithms
Common and Natural Logarithms
 Logarithms can have any base that
you want (or need)
 Common logarithms are on your
calculator and are base 10
 Natural logarithms are also on your
calculator and are base e.
Common Logarithms
103 = 100 is in exponential form this is still a common log
To write it in logarithmic form, write:
log100=3
You do not write a base with a common log, because it is always base 10.
You can also put log100 in the calculator to find the answer 3.
Remember our Intro Page!
A logarithm is an exponent. It also lets you solve equations
that you wouldn’t be able to solve any other way.
For example, you can easily solve
2x = 8, since you know that
23 = 8,
so x = 3.
But what about 2x = 15 ? This is where you would use logs.
Example 1
Solve 2x = 15
1. “DROP LOGS ON IT!”
2. Put the exponent out front
3. Solve using your algebraic rules.
4. Solve in your calculator.
Solve the following problems.
1. 4x = 27
2. 32x = 41
3. 5x = 65
Natural Logarithms
 The graph of
has an
asymptote at 2.7182
 This is the number e. The logarithm
with this base is written as
ln9 =2.1972
Which means that 2.71822.1972=9
Solving problems with base e
 Usually, it doesn’t matter if you use a
common log or a natural log
 If you are using a formula with e,
then you would use a natural log,
since that is the base of the log.
 Ex. The formula for continuous
compounding: A = Pert.
Example 2
If you have $2500 to invest at a rate of 2.5%, how long
would it take to double your money? Assume
continuous compounding.
The formula you would use is : A = Pert
A is the amount you make, P is the original principle invested, e is the
growth factor, r is the rate, and t is the time in years.
Turn in the following problems
1. For a certain credit card, given a starting balance of P and an ending balance of
A, the function
gives the number of months that have passed,
assuming that there were no payments or additional purchases during that time.
a. You started with a debt of $1000 and now owe $1210.26. For how many
months has the debt been building?
b. How many additional months will it take until the debt exceeds $1420?
2. The difference between the apparent magnitude (brightness)m of a star, and
its absolute magnitude M is given by the formula
where d is the distance of the
star from the Earth, measured in parsecs.
a. Find the distance d of Antares from Earth.
b. Sigma Sco is 225 parsecs from Earth. Find its
absolute magnitude.
c. How many times as great is the distance to Antares
as the distance to Rho Oph?