Algebra 2

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Algebra 2

• Section 10-4 Common Logarithms
• Section 10-5 Natural Logarithms

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What You'll LearnWhy 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

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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

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What is the relationship?

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

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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

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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.

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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
• B. log 0.12

Reminder of what a log really is
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Solution Example 1A

• 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
• You can set your calculator to round to 4 decimal
  places

The characteristic is the exponent of 10 when the
original number is expressed in scientific
notation.
2.0792
Mantissa
.0792
Characteristic
120 1.20 x 102
2
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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
• 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

-0.9208
Mantissa
.0792
Characteristic
-1
0.12 1.2 x 10-1
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Example 2

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

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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
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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

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Example 3

• Use a calculator to find the antilogarithm of
  3.073

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Solution Example 3

• Use a calculator to find the antilogarithm of
  3.073

1183
Check
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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

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Example 4

• Use a calculator to find ln 3.925

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Solution Example 4

• Use a calculator to find ln 3.925
• 1.3674

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Antilogarithms

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

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Example 5

• A. Find x if ln x 3.4825
• B. find e if ln e 1

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Solution Example 5A

• A. Find x if ln x 3.4825
• x antiln 3.4825
• x 32.5410

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Solution Example 5B

• B. find e if ln e 1
• e antiln 1
• e 2.7183

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THE END