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 baseOf the exponential form.

The ‘answer’ to the log statement becomes the powerin 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

Types of LogarithmsThere are two special logarithms that your calculatorIs 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.

a. 10x = 75 this transfers to the log statement log 10 75 = x calculator will tell you x = 1.88

b. 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