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4.3.1 Logarithmic Functions
Definition of the Logarithmic Function
Let a be a positive number with a 6 = 1. The logarithmic function with base a, denoted by log (^) a is defined by
Example 1 Express the equation in exponential form. (a) log 3 ( x + 2 ) = 4.
(b) log 5 4 = x. A. 4 x^ = 5 B. x 4 = 5 C. 5 x^ = 4 D. 45 = x E. x 5 = 4
Example 2 Express the equation in logarithmic form. (a) 104 = 10, 000.
(b) e 2 x^ +^1 = 7. A. log 7 ( 2 x + 1 ) = e B. log (^2) x + 1 7 = e C. ln 7 = 2 x + 1 D. log 7 e = 2 x + 1 E. log 7 e = 2 x + 1
Example 3 Evaluate the logarithmic expression. (a) log 2 2
(b) log 5 1
(c) log (^4)
(d) e ln 4
(e) 10 log^
p (^7)
Common Logarithms
The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log 10 x
The logarithm with base e is called the natural logarithm and is denoted by ln:
ln x = log (^) e x
Example 5 Use a calculator to evaluate ln ( 43.5 ) and log ( 85.7 ) , correct to 4 decimal place.
Example 6 Use the definition of the logarithmic function to find x. (a) log 10 0.01 = x
(b) ln x = 1
