Logarithms Worksheets by css39027

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```									   Algebra 2

Section 10-4 Common Logarithms
Section 10-5 Natural Logarithms
What You'll Learn
Why It's Important

• To identify the characteristic and the
mantissa of a logarithm
• To find common logarithms and
antilogarithms
• To find natural logarithms of numbers
• You can use common logarithms to
solve problems involving astronomy
and acoustics.
• You can use natural logarithms to solve
problems involving sales and physics
Common Logarithms

• One of the more useful logarithms
is base 10, because our number
system is base10
• Base 10 logarithms are called
common logarithms
• These are usually written without
subscript 10,
– so log10x is written as log x
What is the relationship?

• Find the logarithms of 5.94, 59.4, 594 and
5940
– What do you notice?
What is the relationship?

• Find the logarithms of 5.94, 59.4, 594 and
5940
– What do you notice?

– The decimal parts are the same, the
integer parts are different
Mantissa & Characteristic

• The decimal part is called the mantissa
– The logarithm of a number between 1
and 10
• The integer part is called the
characteristic
– The integer used to express a base 10
logarithm as the sum of an integer and
a positive decimal
– The characteristic is the exponent of
10 when the original number is
expressed in scientific notation.
Example 1

• Use a scientific or graphing calculator to
find the logarithm for each number
rounded to four decimal places. Then
state the mantissa and characteristic.

• A. log 120
Reminder of
what a log
really is
• B. log 0.12
Solution Example 1A

• Use a scientific or graphing calculator to
find the logarithm for each number rounded
to four decimal places. Then state the     The characteristic is the
exponent of 10 when
mantissa and characteristic.               the original number is
expressed in scientific
• A. log 120 = 2.0792                        notation.

Mantissa = .0792
Characteristic = 2      120 = 1.20 x 102

• You can set your calculator to round to 4
decimal places
Solution Example 1B

• Use a scientific or graphing calculator to find
the logarithm for each number rounded to four
decimal places. Then state the mantissa and
characteristic.
• B. log 0.12 = -0.9208
Mantissa = .0792
Characteristic = -1
0.12 = 1.2 x 10-1
• The mantissa is usually expressed as a
positive number. To avoid negative
mantissas, we rewrite the negative mantissa
as the difference of a positive number and an
integer, usually 10
Example 2

• Use a calculator to find the logarithm
for 0.0038 rounded to four decimal
places. Then state the mantissa and
characteristic.
Solution Example 2

• Use a calculator to find the logarithm for
0.0038 rounded to four decimal places. Then
state the mantissa and characteristic.

Log 0.0038 = -2.4202

Mantissa = 0.5798
Characteristic = -3

0.0038 = 3.8 x 10-3
Antilogarithm

• Sometimes an application of logarithms
requires that you use the inverse of
logarithms, exponentiation.
• When you are given the logarithm of a
number and asked to find the number,
you are finding the antilogarithm.
– That is, if log x = a, then x = antilog a
Example 3

• Use a calculator to find the
antilogarithm of 3.073
Solution Example 3

• Use a calculator to find the antilogarithm
of 3.073

≈1183

Check:
Natural Logarithms

• The number e used in the exponential
growth problem on page 622 is used
extensively in science and
mathematics
• It is an irrational number whose value
is approximately 2.718.
• e is the base for the natural
logarithms, which are abbreviated ln
• The natural logarithm of e is 1
• All properties of logarithms that we
have learned apply to the natural
logarithms as well
• The key marked         on your calculator
is the natural logarithm key
Example 4

• Use a calculator to find ln 3.925
Solution Example 4

• Use a calculator to find ln 3.925

• ≈ 1.3674
Antilogarithms

• You can take antilogarithms of
natural logarithms as well.
• The symbol for the antilogarithm of x
is antiln x
Example 5

• A. Find x if ln x ≈3.4825

• B. find e if ln e = 1
Solution Example 5A

• A. Find x if ln x ≈3.4825
• x ≈ antiln 3.4825

• x ≈ 32.5410
Solution Example 5B

• B. find e if ln e = 1
• e = antiln 1

• e ≈ 2.7183

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