Logarithmic
Functions
Section 6.3
Use the table below.
x
-3
y=10x
-2
-1
1
1
1
10 0 0
10 0
10
0 1
2
3
1 10
100
1000
• Use the table to find x in each equation.
• A. 10x = 1000
• B. 10 x  1
10 0
• C.
10
x
1
• On your graphing calculator:
– 1. Enter the equation y 1  10 x
– 2. See the table of values for this
equation by pressing 2nd table.
– 3. What is 107?
– 4. Now change the table setup so the x
values change by 0.01 by pressing 2nd
table set
– 5. Arrow down to tbl and change to
0.01
– 6. Go back to the table and
x
approximate x for 10  2 5
• To solve an equation such as 10x =
85 or 10x = 2.3 a logarithm is
needed.
• With logarithms you can write an
exponential equation in an equivalent
logarithmic form.
• Exponential form
Logarithmic form
• 103 = 1000
3 = log10 1000
base
exponent
Equivalent exponential and
logarithmic forms
• For any positive base b, where
bx = y if and only if x = logb y
b 1
• Examples: Write each equation in
exponential form.
• 1. log2 8 = 3
8 = 23
• 2. lo g 1   4
1
4
256
 4
256
4
• 3. log39 = 2
• 4. log8512 = 3
• 5. log5625 = 4
32 = 9
83 = 512
54 = 625
• Write each equation in logarithmic
form.
• 1. 153 = 3375
log153375 = 3
• 2. 4  2
log42 = ½
• 3. 53 = 125
log5125 = 3
• 4. 2 7  3
log273= 1/3
• 5. 113 = 1331
log111331 = 3
1
2
1
3
Evaluate logarithmic expressions using
your calculator
• You can evaluate logarithms with a
base of 10 by using the log key on a
calculator.
• Find the approximate value of each
logarithmic function. Round to the
nearest tenth.
• 1. log10 870 = 2.9
• 2. log1098,560 = 5.0
• 3. log10.0000056 = -5.3
• Solve 10x = 85 for x. Round to the
nearest hundredth.
• Write the equation in logarithmic form and
use the log key.
• x = log10 85  1.93
• Solve 10x = 14.5 for x. Round to the
nearest hundredth.
• x = log10 14.5 x  1 .1 6
•
Day 2
• Evaluate without a calculator.
• Ex 1 lo g3 9
• Ex. 2 log10 10 0 0
• Ex. 3 log 1
4
• Ex. 4
log11
16
1
12 1
Day 2
• Case 2: Solving using rational
exponents
• When: Use when the variable is the
base of an exponential expression.
• How: Raise both side of the
equation to the reciprocal of the
power of the exponent.
Examples.
• Ex. 1
4 w
4
3
Ex 2.
1
3
m 2  64
3

 w

64  w
1
3



3
2
2
 23  3
3
m

6
4






m  16
• Ex. 3 1
64
1
64
1
4
64 3

n

3
Ex. 4
4
2 16
1
1
3
n
2 16
4

n

256  n
3
4
 x
4
3


 216  3
2
3
x
2
3
1

1
1

1

2
1

2
 23  3
x 


x  36
• Case 3: One-to-One Property of
Exponents: If bx = by, then x = y
• When to use: bases are the same
number or can be changed to the
same number
• How: set exponents equal.
• Note: will be on non-calculator
portion quiz/test
• Ex. 1
5
x
Ex. 3
7
5
3
x 7
• Ex. 2
2
x
8
2
x
 2
2
x
2
x 9
9
3
3

9
3
3
2 x 1
2

3
2 x 1
4 x 2
6x
3
3
6x
6x
4 x  2  6x
 2  2x
x  1
Solve for v.
• Write in exponential form and solve.
• Ex. v = log125 5
• 125v = 5
• (53)v = 5
• 53v = 51
• 3v = 1
• v = 1/3
• Ex. 2v = 1
•v=0
• (remember: any base raised to the
zero power = 1)