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Logarithms Worksheets document sample
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