LOGARITHMS

LOGARITHMS

1.    Definitions

The logarithm of a number A to base [B ¹ 0 or 1] is the power to which the base must be raised to equal A.

2.    Shorthand notation

log A = log10 A = the logarithm of A to base 10; ln A = loge A = the natural logarithm of A

3.    Rules [A, B ¹ 0 or 1 and c ¹ 0]

A = BlogBA = logB(BA)

logc(AB) = logcA + logc B

logc(A/B) = logcA - logc B

logcAx = x logc A

logc(1/A) = - logc A

4.    Examples

 log1000 = 3 log 10 = 1 log 1 = 0 log 1/100 = -2 log 0 = - ¥ log 2 = .3010 log 3 = .4771 log 5 = .6890 log 7 = .8451 log 1.5 = .1761 log (-5) = undefined

5.    logarithm of any  number

log 105.6 = log(100 x 1.056) = log 100   +   log 1.056 = 2   +   log 1.056

log 0.2 = log 2/10 = log 2 - log 10 = .3010 - 1

So, for base 10, need to know logarithms from 1 to 10.

6.    Uses

§         Polynomials [e.g. 5x4 + 3x2 + 2x +1], logarithms, exponentials, and the trigonometric functions are the “elementary functions of analysis” that provide essential skills to be able to use calculus and other parts of higer mathematics effectively.

§         Multiplication before calculators

§         Understanding and detecting exponential growth. Examples of exponential growth are cells in a colony that double every half hour, or interest where a bank balance that grows by the same multiple every year. Exponential decay is similar: the concentration of many medications and drugs in the blood stream halves in a period of time called the "half-life." A graph of an exponential function soon goes “off the paper” but a graph of the log of the exponential function is a straight line. This idea is also used to detect exponential behavior: if the graph of the log of something is a straight line then the growth or decay is exponential.

7.    Any base

A = ClogCA

Take log base B:

logBA = logCA x logBC

logCB = 1/logBC

logBA = logCA/logCB