Logarithmic Functions Section 5.5: Logarithmic Functions

log x = a if and only if 10a = x common logarithm – the common logarithm of any positive real number x is defined to be the exponent you get when you write x as a power of 10. log x = a if and only if 10a = x

The important thing to remember is the log represents the exponent.  In the case of common logs, the base is always base 10.  Study the following examples. 1)  log 100 = 2 because 102 = 100. 2)  log 1000 = 3 because 103 = 1000. 3)  log 1 = 0 because 100 = 1. 4)  log .1 = -1 because 10-1 = .1 5)  log .01 = -2 because 10-2 = .01

The log function is the inverse function of the exponential function and as such their graphs are reflections about the y = x line.  Here is the graph of the common log and the inverse.

Some important facts you need to understand from the log graph.  The domain of the log is x > 0.  The range is all real numbers.  The zero is at x = 1.  You can only find the log of positive numbers.  Logs of numbers less than one are negative and logs of numbers greater than one are positive.

logb x = n if and only if x = bn We can find the log of other bases by using the following formula similar to the common log definition. logb x = n if and only if x = bn Here are some examples: 1) log2 8 = 3 because 23 = 8 2)  log3 81 = 4 because 34 = 81. 3)  log4 1/16 = -2 because 4-2 = 1/16 4) log8 1 = 0 because 80 = 1

One of the most important log function is called the natural log which has the number e as the base.  When e is used as a base we use the following notation: ln x = a if and only if ea = x Most natural logs need to be calculated on your calculator.  The graph of the natural log is shown on the next slide.

Example 1: Evaluate. a) log82 Let the log82 = x 8x = 2 (23)x = 21 23x = 21 3x = 1 x = 1/3

ln 1 e3 ln 1 = x ex = 1 ex = e-3 x = -3

log 1 . 10,000 x = -4 log51 = x x = 0

Example 2: Solve. a) log x = 4 x = 104 x = 10,000

ln x = ½ e1/2 = x x ≈ 1.65 c) log x = -1.2 10-1.2 = x x ≈ 0.063

HOMEWORK p. 194 – 195; 12 – 20 even