Logarithmic Functions
Section 3-2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
For x 0 and 0 a 1, y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the
logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
A logarithm is an exponent!
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
BIG PICTURE
• Logarithms are just another way to write an exponential expression
• Cannot take Log of ZERO or any NEGATIVE #
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
Exponential Equation
nb p
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Logarithmic Equation
pnb log0n
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
Conversion between Formats
nb p pnb log
pnb log
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
Conversions
• Logarithmic Equations
• log5 25 = 2
• log8 1 = 0
• log9 3 = ½
• log10 100,000 = 5
• Exponential Format
• 52 = 25
• 80 = 1
• 9½ = 3
• 105 = 100,000
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
log10 –4 LOG –4 ENTER ERROR
no power of 10 gives a negative number
The base 10 logarithm function f (x) = log10 x is called the
common logarithm function.
The LOG key on a calculator is used to obtain common logarithms.
Examples: Calculate the values using a calculator.
log10 100
log10 5
Function Value Keystrokes Display
LOG 100 ENTER 2
LOG 5 ENTER 0.69897005
2log10( ) – 0.3979400LOG ( 2 5 ) ENTER
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log7
9
7log7
9 = 9 property 3
Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a.
4. If loga x = loga y, then x = y. one-to-one property 3. loga a
x = x and alogax = x inverse property
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
x
y
Graph f (x) = log2 x
Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
83
42
21
10
–1
–2
2xx
4
1
2
1
y = log2 x
y = xy = 2x
(1, 0)
x-intercept
horizontal asymptote y = 0
vertical asymptote x = 0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
Example: Graph the common logarithm function f(x) = log10 x.
by calculator
1
10
10.6020.3010–1–2f(x) = log10 x
10421x 1
100
y
x5
–5
f(x) = log10 x
x = 0 vertical
asymptote
(0, 1) x-intercept
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)
3. x-intercept (1, 0)
5. increasing6. continuous7. one-to-one 8. reflection of y = a
x in y = x
1. domain ),0( 2. range ),(
4. vertical asymptote )(0 as 0 xfxx
Graph of f (x) = loga x (a 1)
x
y y = x
y = log2 x
y = a x
domain
range
y-axisvertical
asymptote
x-intercept(1, 0)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
The function defined by f(x) = loge x = ln x
is called the natural logarithm function.
Use a calculator to evaluate: ln 3, ln –2, ln 100
ln 3ln –2ln 100
Function Value Keystrokes Display
LN 3 ENTER 1.0986122ERRORLN –2 ENTER
LN 100 ENTER 4.6051701
y = ln x
(x 0, e 2.718281)
y
x5
–5
y = ln x is equivalent to e y = x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
2
1ln
e 2ln 2 e inverse property
20lne 20 inverse property
eln3 3)1(3 property 2
00 1ln property 1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
Log Functions Overview
• Log Function Base
• loga x a
• log x 10
• loge x e
• ln x e
16
Homework
• WS 6-2
• Quiz next class on graphing