1. Warm-up: Solve for x. 2x = 8 2) 4x = 1 3) ex = e 4) 10x = 0.1
HW: Page 317 (2 – 18 even, 21, 22, 25, 26, 29, 30, even, 51 – 61 odd)
Warm-up:
Solve for x.
2x = 8 2) 4x = 1 3) ex = e 4) 10x = 0.1
Describe how the graph of g is related to the graph of f.
5) g(x) = f(x + 2) 6) g(x) = -1 + f(x)
7) g(x) = -f(x) 8) g(x) = f(-x)

2. 3.2 Logarithmic Functions and Their Graphs
Objective:
Graph logarithmic and natural log functions
Apply properties of logarithms and natural logarithms

3. y = loga x if and only if x = a y.
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 logarithm is an exponent!
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay

4. Equivalent Exponential Equation
Examples: Evaluate each Logarithm.
y = logax is equivalent to x = ay
Solution
Equivalent Exponential Equation
Logarithm
log216
16 = 2y
16 = 24  y = 4
log2( )
= 2 y
= 2-1 y = –1
log1255
5 = 125y
5 = 1251/3  y = 1/3
log51
1 = 5y
1 = 50  y = 0

5. 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.
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

6. 3. loga ax = x and alogax = x inverse property
Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x and alogax = x inverse property
4. If loga x = loga y, then x = y. one-to-one property
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log79
7log79 = 9 property 3

7. horizontal asymptote y = 0
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 y = 2x in the line y = x. x y
y = 2x
y = x 8 3 4 2 1 –1 –2 2x x
horizontal asymptote y = 0
y = log2 x
x-intercept
(1, 0)
vertical asymptote
x = 0

8. Example: Graph the common logarithm function f(x) = log10 x.
0.602
0.301 –1 –2
f(x) = log10 x 10 4 2 x y x 5 –5
by calculator
f(x) = log10 x
(1, 0) x-intercept
x = 0 vertical asymptote

9. The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a  1)
y-axis
vertical
asymptote x y
Graph of f (x) = loga x (a  1)
y = a x
y = x
range
1. domain
y = log2 x
2. range
3. x-intercept (1, 0)
4. vertical asymptote
domain
x-intercept
(1, 0)
5. increasing
6. continuous
7. one-to-one
8. reflection of y = a x in y = x

10. Ex: Graph the logarithmic function f(x) = log10(x – 1)
SHIFT f(x) = log10x to the RIGHT 1 y x 5 –5
f(x) = log10 (x – 1)
(2, 0) x-intercept
x = 1 vertical asymptote

11. Ex: Graph the logarithmic function f(x) = 2 + log10x
SHIFT f(x) = log10x to the UP 2
f(x) = 2 + log10x y x 5 –5
(1, 2)
x = 0 vertical asymptote

12. The function defined by f(x) = loge x = ln x5 –5
y = ln x
The function defined by f(x) = loge x = ln x
(x  0, e )
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

13. 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.
inverse property
inverse property
property 2
property 1

14. Finding the Domains of Logarithmic Functions
Every domain of every ln x is the set of positive real numbers.
(ln (-1)) is undefined.
a. f(x) = ln(x – 2) b. g(x) = ln(2 – x) c. h(x) = lnx2
x – 2 > 0
2 – x > 0
x2 > 0
D: (2, )
D: (-, 2)
D: All Reals except x = 0

15. Sneedlegrit:
Find the domain, vertical asymptote, and x-intercept of f(x) = log4(x – 3) then sketch its graph.
HW: Page 317 (2 – 18 even, 21, 22, 25, 26, 29, 30, even, 51 – 61 odd)