8.4 Logarithmic Functions

What is a Logarithm? We know 22 = 4 and 23 = 8, but for what

value of x does 2x = 6?It must be between 2 and 3…Logarithms were invented to solve

exponential equations like this.x = log26 ≈ 2.585

Logarithms with Base bLet b and y be positive numbers and b≠1.The logarithm of y with base b is written

logby and is defined:

logby = x if and only if bx = y

Rewriting Log EquationsWrite in exponential form:

log2 32 = 5

log5 1 = 0

log10 10 = 1

log10 0.1 = -1

log1/2 2 = -1

Special Log ValuesFor positive b such that b ≠ 1:Logarithm of 1: logb 1 = 0 since b0 = 1

Logarithm of base b: logb b = 1 since b1 = b

Evaluating Log ExpressionsTo find logb y, think “what power of b will

give me y?”Examples:log3 81

log1/2 8

log9 3

Your Turn!Evaluate each expression:

log4 64

log32 2

Common and Natural LogsCommon Logarithm - the log with base

10Written “log10” or just “log”

log10 x = log x

Natural Logarithm – the log with base e Can write “loge“ but we usually use “ln”

loge x = ln x

Evaluating Common and Natural LogsUse “LOG” or “LN” key on calculator.Evaluate. Round to 3 decimal places.log 5ln 0.1

Evaluating Log FunctionsThe slope s of a beach is related to the

average diameter d (in mm) of the sand particles on the beach by this equation:

s = 0.159 + 0.118 log d

Find the slope of a beach if the average diameter of the sand particles is 0.25 mm.

InversesThe logarithmic function g(x) = logb x

is the inverse of the exponential function f(x) = bx.

Therefore:

g(f(x)) = logb bx = x and f(g(x)) = blogb x = xThis means they “undo” each other.

Using Inverse PropertiesSimplify:

10logx

log4 4x

9log9 x

log3 9x

Your Turn!Simplify:

log5 125x

5log5 x

Finding InversesSwitch x and y, then solve for y.Remember: to “chop off a log” use the

“circle cycle”!Find the inverse:

y = log3 x y = ln(x + 1)

Your Turn!Find the inverse.y = log8 x

y = ln(x – 3)

Logarithmic GraphsRemember f-1 is a reflection of f over the

line y = x.Logs and exponentials are inverses!

exp. growth exp. decay

Properties of Log GraphsGeneral form: y = logb (x – h) + kVertical asymptote at x = h.

(x = 0 for parent graph)Domain: x > hRange: All real #sIf b > 1, graph moves up to the rightIf 0 < b < 1, graph moves down to the

right.

To graph:Sketch parent graph (if needed).

Always goes through (1, 0) and (b, 1)Choose one more point if needed.Don’t cross the y-axis!

Shift using h and k.Be Careful: h is in () with the x, k is not

Examples:Graph. State the domain and range.

y = log1/3 x – 1

Domain:

Range:

Graph. State the domain and range.

y = log5 (x + 2)

Domain:

Range:

Your Turn!Graph. State the domain and range.

y = log3 (x + 1)

Domain:

Range: