A logarithm is an exponent.
In the example shown at the right, 3 is the exponent to which the
base 2 must be raised to create the answer of 8, or 23 = 8.
In this example, 8 is called the antilogarithm base 2 of 3.
logpic
Try to remember the "spiral" relationship between the values as shown at the right. Follow the arrows starting with base 2 to get the equivalent exponential form, 23 = 8.
logrule

bullet A logarithm base b of a positive number x is such that:
for b > 0, b≠ 1, logb x = y if and only if by = x.
The log bx is read "log base b of x".
The logarithm y is the exponent to which b must be raised to get x.

Logarithms with base 10 are called common logarithms. When the base is not indicated, base 10 is implied.
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Logarithms with base e are called
natural logarithms. Natural logarithms
are denoted by ln.
log28

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Properties of Logarithms

Using the properties of "exponents", we can arrive at the properties of "logarithms".

Properties of
Exponents:

log30
Let's find the connection!
log31
Similar investigations lead to the other logarithm properties.
Properties of Logarithms:

log125
log126
logr
Also, log44

These log properties remain the same when
working with the natural log, ln(x).
orangearrow
Remember:    ln 1 = 0    and    ln e = 1
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Examples:

1.
 Write le1 in exponential form. Solution: le1a
2.
 Write le2in logarithmic form. Solution: le2a
3.
 Evaluate: le3 Solution: le3a
Yes, you could use your calculator.
4.
 What is the value of x?   le4 Solution: le4a
5.
 Write in expanded form: le5
(Apply the "Properties of Logs" rules.)		 Solution: le5a
6.
 Write in expanded form: le6 Solution:le6a
7.

Express as a single logarithm:le7

(Apply the "Properties of Logs" rules in reverse)

 Solution: le7a
8.
 Express as a single logarithm:le8
 Solution: le8a
9.
 Using "Properties of Logarithms", show thatle9 Solution:
le9a
10.
 Using "Properties of Logs", solve for x.le10
 Solution:
le10a

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Change of Base Formula:log97
There is a property called the Change of Base Formula that exists regarding logarithms.
This formula is used with older models of calculators that cannot accept subscripted typing on the screen. While a key is present for log base 10, this formula is needed for log entries with other bases.

Let's take a look at how this formula was created, using "Properties of Logarithms":

Origins of Change of Base Formula:
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Set = x.
l0g21
Convert to exponential form.
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Take common log of both sides.
log23
User power rule.
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Divide by log b.





How to use your graphing calculator for working
with
logarithms
Click here.
ti84c
How to use
your graphing calculator for
working
with
logarithms
 Click here.


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