logarithms

LOGARITHMS

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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 = log_e A = the natural logarithm of A

3. Rules [A, B ¹ 0 or 1 and c ¹ 0]

A = B^log_B^A = log_B(B^A)

log_c(AB) = log_cA + log_c B

log_c(A/B) = log_cA - log_c B

log_cA^x = x log_c A

log_c(1/A) = - log_c 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 = C^log_C^A

Take log base B:

log_BA = log_CA x log_BC

log_CB = 1/log_BC

log_BA = log_CA/log_CB