Pre-Calc Lesson 5-5 Logarithms Logarithmic functions--- are the inverse of the exponential function Basic exponential function: f(x) = bx Basic logarithmic function: f-1(x) = logbx Every (x,y)  (y,x)

Basic rule for changing exponential equations to logarithmic equations (or vice-versa): logbx = a  ba = x The base of the logarithmic form becomes the base Of the exponential form. The ‘answer’ to the log statement becomes the power in the exponential form . The number you are to take the ‘log’ of in the log form, becomes the answer in the exponential form.

Examples: log525 = 2 because 52 = 25 log5125 = 3 because 53 = 125 log2(1/8) = - 3 because 2-3 = 1/8 base ‘b’ exponential function f(x) = bx Domain: All reals Range: All positive reals base ‘b’ logarithmic function f-1(x) = logb(x) Domain: All positive reals Range: All reals

There are two special logarithms that your calculator Types of Logarithms There are two special logarithms that your calculator Is programmed for: log10(x)  called your ‘common logarithm’ For the common logarithm we do not include the subscript 10, so all you will see is: log (x) So log10(x)  log (x) = ‘k’ iff 10k = x loge(x)  called your ‘natural logarithm’ For the natural logarithm we do not include the subscript e, so all you will see is: ln (x) So loge(x)  ln (x) = ‘k’ iff ek = x

log 6.3 = 0.8 because 100.8 = 6.3 ln 5 = 1.6 because e1.6 = 5 Example 2 : Find the value of ‘x’ to the nearest hundredth. 10x = 75 this transfers to the log statement log 10 75 = x calculator will tell you  x = 1.88 ex = 75 this transfers to the log statement ln 75 = x calculator will tell you  x = 4.32

Examples: Evaluate: a. log 8 2 b. ln 1 e3 c. log 1 . 10,000 d. log 5 1 Solve: a. log x = 4 b. ln x = 1 2 c. log x = - 1.2