1. 3.2 Logarithmic Functions

2. Solving for an exponent
Intro
Solving for an answer
Solving for a base
Solving for an exponent
1. 34 = x
2. x2 = 49
3. 3x = 729
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

3. Now you try…
4. 25 = x 5. x3 = x = 3125
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

4. Definition: Logarithmic Function
A logarithmic function is the inverse function of an exponential function.
Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithm is an exponent!
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Definition: Logarithmic Function

5. Examples: Write Equivalent Equations
Examples: Write the equivalent exponential equation
and solve for y.
Solution
Equivalent Exponential Equation
Logarithmic Equation
y = log216
16 = 2y
16 = 24  y = 4
y = log2( )
= 2 y
= 2-1 y = –1
y = log416
16 = 4y
16 = 42  y = 2
y = log51
1 = 5 y
1 = 50  y = 0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Examples: Write Equivalent Equations

6. baseexp =ans  logbaseans=exp
3.2 Material
Logarithms are used when solving for an exponent.
baseexp =ans  logbaseans=exp
In your calculator: log is base ln is base e
Properties of Logarithms and Ln
loga 1 = 0
logaa = 1
logaax = x
The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

7. Common Logarithmic Function
Examples: Calculate the values using a calculator.
Function Value
Keystrokes
Display
log10 100
LOG 100 ENTER 2
log10( )
LOG ( ) ENTER –
log10 5
LOG ENTER
log10 –4
LOG –4 ENTER
ERROR
no power of 10 gives a negative number
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Common Logarithmic Function

8. Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x
4. If loga x = loga y, then x = y. one-to-one property
Examples:
1. Solve for x: log6 6 = x
2. Simplify: log3 35
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Properties of Logarithms

9. Graphs of Logarithmic Functions
The graphs of logarithmic functions are similar for different values of a f(x) = loga x
y-axis
vertical
asymptote x y
Graph of f (x) = loga x
y = a x
y = x
y = log2 x
x-intercept (1, 0)
VA: x = 0
increasing
x-intercept
(1, 0)
reflection of y = a x in y = x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Graphs of Logarithmic Functions

10. Example: Graph the common logarithm function f(x) = log10 x.y x 5 –5
f(x) = log10 x
(1, 0) x-intercept
x = 0 vertical asymptote
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: f(x) = log0 x

11. Natural Logarithmic Functiony x 5 –5
y = ln x
The function defined by f(x) = loge x = ln x
is called the natural logarithm function.
y = ln x is equivalent to e y = x
Use a calculator to evaluate: ln 3, ln –2, ln 100
Function Value
Keystrokes
Display
ln 3
LN 3 ENTER
ln –2
LN –2 ENTER
ERROR
ln 100
LN 100 ENTER
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Natural Logarithmic Function

12. Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
inverse property
property 2
property 1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Properties of Natural Logarithms

13. Classwork
Pg 236 # 1-21 odd
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

14. Example: Carbon Dating
Example: The formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is How old is it?
original equation
multiply both sides by 1012
take the ln of both sides
inverse property
To the nearest thousand years the charcoal is 57,000 years old.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Carbon Dating

15. 3.2 Material logab=c ac =b
Logarithms are used when solving for an exponent.
logab=c ac =b
In your calculator: log is base ln is base e
Properties of Logarithms and Ln
loga 1 = 0
logaa = 1
logaax = x
The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

16. Practice Problems Page 236 #27–29, 32-38, 80-86 even
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.