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.
log27
Logarithms with base e are called
natural logarithms. Natural logarithms
are denoted by ln.
log28

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hintgal
Logarithms with base 10 are called common logarithms and are written without the 10 showing.
The log key will calculate common
(base 10) logarithms.
logkey
Logarithms with the base e are called natural logarithms and are written using the notation ln(x).
The ln key will calculate natural
(base e) logarithms.
lnkey
For "other bases" use the change of base formula:
logcalcformula
log2f
is entered aslog22g

Origins of Change of Base Formula:
log21
Set = x.
l0g21
Convert to exponential form.
log22
Take common log of both sides.
log23
User power rule.
log24
Divide by log b.
Change of Base Formula:log97
Calculator Entry:

log17    arrowred   logpic

 

ti84c

For help with logarithms on your calculator, click here.



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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.
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Remember:    ln 1 = 0    and    ln e = 1
log127

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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
If using your calculator, remember to use the change of base formula and enter log 1 / log 3.
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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