4-3: Common and 4-3: Common and Natural LogarithmsNatural Logarithms4-3: Common and 4-3: Common and Natural LogarithmsNatural Logarithms

English CasbarroEnglish CasbarroUnit 4: Exponents/LogarithmsUnit 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.

103 = 100 is in exponential form this is still a common logTo 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.

Common Logarithms

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.

Remember our Intro Page!

Example 1

1. “DROP LOGS ON IT!”

2. Put the exponent out front

3. Solve using your algebraic rules.

4. Solve in your calculator.

Solve 2x = 15

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 thegrowth factor, r is the rate, and t is the time in years.

Turn in the following problems1. 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?