LOGARITHMS
ANIL MITRA PHD, COPYRIGHT © 2000, REFORMATTED June 25, 2003
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 = log_{10} 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_{c}A + log_{c} B
log_{c}(A/B) = log_{c}A  log_{c} B
log_{c}A^{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. 5x^{4} + 3x^{2} + 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 "halflife." 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_{B}A = log_{C}A x log_{B}C
log_{C}B = 1/log_{B}C
log_{B}A = log_{C}A/log_{C}B
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