Greenwood press
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LINEAR FUNCTIONS LOGARITHMS Logarithms are exponents, so they are used to reduce very large values into smaller, more manageable numbers. It is easier to refer to the number 13.4 than the number 25,118,900,000,000, which is approximately equal to 10 13.4 . A num- ber x is said to be the base b logarithm of a number y, if y = b x . The correspon- ding logarithmic equation is x = log b y. Base-10 logarithms are used to change numbers to powers of 10. For example, 500 ≈ 10 2.69897 , so 2.69897 is said to be the base-10 logarithm of 500. This is commonly written as log 500 ≈ 2.69897. The decimal part “.69897” is called the mantissa, and the integer part “2” is called the characteristic. Until inexpensive calculators made it easy to do multi- plication, division, and roots, scientists and engineers used base-10 logarithms to simplify computations by changing multiplication of numbers into addition of exponents, and division of numbers into subtraction of exponents. Up until twenty years ago, the main computational device for high school students in advanced math and sciences was based on logarithmic scales—the slide rule. Other common bases for logarithms are the numbers e and 2. The number e ≈ 2.718281828459. It can be developed from the compound-interest formula as the limit of (1 + 1/n) n as n increases without bound. The base e is used in exponential expressions that evaluate continuously compounded interest. Logarithms to the base e are typically written with the abbreviation ln, called a natural logarithm. ln(500) ≈ 6.21461, because 500 ≈ e 6.21461 . Mathematical functions using e and ln simplify computations with rates and areas that result from situations in physics, biology, medicine, and finance. Hence e and natural logarithms are often used in the statement of rules and properties in these fields. Base-2 logarithms emerge from the study of computer algorithms. Computers are based on on-off switches, so using base-2 logarithms provides a natural connection with machine operations. Logarithmic scales are used in newspapers, households, and automobiles as well as in scientific research. How loud is a rock concert? Noise is measured in decibels, a logarithmic scale that is easier to use than the sound-energy measure- ment of watts per square meter. A decibel is one-tenth of a bel, a unit named after Alexander Graham Bell (1847–1922), inventor of the telephone. A soft whisper is 30 decibels. Normal conversation is at 60 decibels. If you are close to the stage at a rock concert, you hear music at 120 decibels. If you are so close that the music hurts your ears, the amplifiers are at 130 decibels. Because the decibel scale is logarithmic, changes along the scale are not linear. When the rock music moves from very loud (120 decibels) to painful (130 decibels), your ears are receiving 10 times as much sound energy. The difference of 70 decibels between normal conversation (60 decibels) and pain (130 decibels) represents 10 7 more watts per square meter of sound energy. People’s perceptions of changes in sound intensity are more aligned to the decibel scale rather than the actual changes in energy level. The same goes for the perception of light. The brightness of stars was first put on a quantitative scale by the Greek astronomer Hipparchus at around 130 BC . He arranged the vis- ible stars in order of apparent brightness on a scale that ran from 1 to 6 magni- Download 1.81 Mb. Do'stlaringiz bilan baham: |
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