8.4 logarithmic functions
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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: