MATHEMATICS: The exponent, or power, to which 10 has to be raised to express any positive real number.
Logarithm is derived from Greek logos "reckoning, ratio," and arithmos "number."
Logarithm is derived from Greek logos "reckoning, ratio," and arithmos "number."
Since I can't make a nice table, let's use the following format: Base, Exponent, Expression, Result such that in line 1, Base = 10, Exponent = -3, Expression = 10^-3, Result = 0.001. We obtain,
10, -3,10^-3, 0.001 (or 1/1000) (line 1)
10, -2, 10^-2, 0.01 (or 1/100)
10, -1, 10^-1, 0.1 (or 1/10)
10, 0, 10^0, 1
10, 1, 10^1, 10
10, 2, 10^2, 100 (10 squared)
10, 3, 10^3, 1,000 (10 cubed)
And so forth.
Any positive real number can be expressed as the product of 10 raised to any real number; for example 100,000 can be written as 100 x 1000 = 10^2 x 10^3 = 10^5. Notice that the exponents are additive. It is easy to show that for division the exponents subtract.
Before the advent of hand-held electronic calculators, logarithms and the use of log tables reduced calculating time by converting long-hand multiplication into an addition process and long-hand division into a subtraction process where the result was accurate to three significant figures. One would just look up the logarithms of two or more numbers that were being multiplied, sum the logarithms, and then look up the corresponding number.
Another benefit of using logarithms is that curvilinear data points can be converted into linear data points, and the latter is easier to model with a first-order equation derived using either graph paper or linear regression analysis.
10, -3,10^-3, 0.001 (or 1/1000) (line 1)
10, -2, 10^-2, 0.01 (or 1/100)
10, -1, 10^-1, 0.1 (or 1/10)
10, 0, 10^0, 1
10, 1, 10^1, 10
10, 2, 10^2, 100 (10 squared)
10, 3, 10^3, 1,000 (10 cubed)
And so forth.
Any positive real number can be expressed as the product of 10 raised to any real number; for example 100,000 can be written as 100 x 1000 = 10^2 x 10^3 = 10^5. Notice that the exponents are additive. It is easy to show that for division the exponents subtract.
Before the advent of hand-held electronic calculators, logarithms and the use of log tables reduced calculating time by converting long-hand multiplication into an addition process and long-hand division into a subtraction process where the result was accurate to three significant figures. One would just look up the logarithms of two or more numbers that were being multiplied, sum the logarithms, and then look up the corresponding number.
Another benefit of using logarithms is that curvilinear data points can be converted into linear data points, and the latter is easier to model with a first-order equation derived using either graph paper or linear regression analysis.
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