Rewrite logarithmic equations

1
EXAMPLE 1
Rewrite logarithmic equations
Logarithmic Form
Exponential Form
2
for Example 1
GUIDED PRACTICE
Rewrite the equation in exponential form.
Logarithmic Form
Exponential Form
3
EXAMPLE 5
Use inverse properties
Simplify the expression.
SOLUTION
Express 25 as a power with base 5.
Power of a power property
4
EXAMPLE 6
Find inverse functions
Find the inverse of the function.
SOLUTION
b.
y ln (x 3)
Write original function.
x ln (y 3)
Switch x and y.
Write in exponential form.
Solve for y.
5
for Examples 5 and 6
GUIDED PRACTICE
Simplify the expression.
SOLUTION
SOLUTION
6
for Examples 5 and 6
GUIDED PRACTICE
Simplify the expression.
SOLUTION
Express 64 as a power with base 2.
Power of a power property
SOLUTION
eln20
7
for Examples 5 and 6
GUIDED PRACTICE
SOLUTION
SOLUTION
y ln (x 5)
Write original function.
x ln (y 5)
Switch x and y.
Write in exponential form.
Solve for y.
8
EXAMPLE 7
Graph logarithmic functions
Graph the function.
SOLUTION
Plot several convenient points, such as (1, 0),
(3, 1), and (9, 2). The y-axis is a vertical
asymptote.
From left to right, draw a curve that starts just
to the right of the y-axis and moves up through
the plotted points, as shown below.
9
EXAMPLE 7
Graph logarithmic functions
Graph the function.
SOLUTION
Plot several convenient points, such as (1, 0),
(2, 1), (4, 2), and (8, 3). The y-axis is a
vertical asymptote.
From left to right, draw a curve that starts just
to the right of the y-axis and moves down through
the plotted points, as shown below.
10
EXAMPLE 8
Translate a logarithmic graph
SOLUTION
STEP 1
STEP 2
Translate the parent graph left 3 units and up 1
unit. The translated graph passes through (2,
1), (1, 2), and (1, 3). The graphs asymptote is
x 3. The domain is x gt 3, and the range is
all real numbers.
11
for Examples 7 and 8
GUIDED PRACTICE
Graph the function. State the domain and range.
SOLUTION
12
for Examples 7 and 8
GUIDED PRACTICE
From left to right, draw a curve that starts just
to the right of the y-axis and moves up through
the plotted points.
The domain is x gt 0, and the range is all real
numbers.
13
for Examples 7 and 8
GUIDED PRACTICE
Graph the function. State the domain and range.
SOLUTION
domain x gt 3, range all real numbers
14
for Examples 7 and 8
GUIDED PRACTICE
Graph the function. State the domain and range.
SOLUTION
domain x gt 21, range all real numbers
15
EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm.
SOLUTION
16
EXAMPLE 2
Evaluate logarithms
Evaluate the logarithm.
SOLUTION
17
EXAMPLE 3
Evaluate common and natural logarithms
Expression
Keystrokes
Display
Check
0.903089987
1.203972804
18
EXAMPLE 4
Evaluate a logarithmic model
19
EXAMPLE 4
Evaluate a logarithmic model
SOLUTION
Write function.
93 log 220 65
Substitute 220 for d.
Use a calculator.
282.806
Simplify.
20
for Examples 2, 3 and 4
GUIDED PRACTICE
Evaluate the logarithm. Use a calculator if
necessary.
SOLUTION
2 to what power gives 32?
SOLUTION
27 to what power gives 3?
21
for Examples 2, 3 and 4
GUIDED PRACTICE
Evaluate the logarithm. Use a calculator if
necessary.
Expression
Keystrokes
Display
Check
1.079
0.288
22
for Examples 2, 3 and 4
GUIDED PRACTICE
SOLUTION
Write function.
93 log 150 65
Substitute 150 for d.
Use a calculator.
267
Simplify.
23
EXAMPLE 1
Use properties of logarithms
Quotient property
Simplify.
Product property
Simplify.
24
EXAMPLE 1
Use properties of logarithms
Power property
Simplify.
25
for Example 1
GUIDED PRACTICE
Quotient property
Simplify.
Product property
Simplify.
26
for Example 1
GUIDED PRACTICE
Power property
Simplify.
Power property
Simplify.
27
EXAMPLE 2
Expand a logarithmic expression
SOLUTION
Quotient property
Product property
Power property
28
EXAMPLE 3
Standardized Test Practice
SOLUTION
Power property
Product property
Quotient property
Simplify.
29
for Examples 2 and 3
GUIDED PRACTICE
SOLUTION
Product property
Power property
30
for Examples 2 and 3
GUIDED PRACTICE
SOLUTION
ln 4 3 ln 3 ln 12
Power property
Product property
Quotient property
Simplify.
31
EXAMPLE 4
Use the change-of-base formula
SOLUTION
Using common logarithms
Using natural logarithms
32
EXAMPLE 5
Use properties of logarithms in real life
For a sound with intensity I (in watts per square
meter), the loudness L(I) of the sound (in
decibels) is given by the function
33
EXAMPLE 5
Use properties of logarithms in real life
SOLUTION
Let I be the original intensity, so that 2I is
the doubled intensity.
Increase in loudness
Write an expression.
Substitute.
Distributive property
Product property
Simplify.
Use a calculator.
34
for Examples 4 and 5
GUIDED PRACTICE
Use the change-of-base formula to evaluate the
logarithm.
SOLUTION
SOLUTION
35
for Examples 4 and 5
GUIDED PRACTICE
Use the change-of-base formula to evaluate the
logarithm.
SOLUTION
SOLUTION
36
for Examples 4 and 5
GUIDED PRACTICE
SOLUTION
Let I be the original intensity, so that 3I is
the tripled intensity.
37
for Examples 4 and 5
GUIDED PRACTICE
Write an expression.
Substitute.
Distributive property
Product property
Simplify.
Use a calculator.