Logarithmic function, equation and inequality

LOGARITHMIC FUNCTION,
LOGARITHMIC EQUATION AND
LOGARITHMIC INEQUALITY
 FELINA E. VICTORIA
 MAED-MATH

Example 1:
Solution: log2 8  3
We read this as: ”the log
base 2 of 8 is equal to 3”.
3
Write 2 8 in logarithmic form.

Example 1a:
Write 42
16 in logarithmic form.
Solution: log4 16  2
Read as: “the log
base 4 of 16 is
equal to 2”.

Example 1b:
Solution:
Write 2
 3

1
8
in logarithmic form.
log2
1
8
  3
1
Read as: "the log base 2 of is equal to -3".
8

Okay, so now it’s time for
you to try some on your
own.
1. Write 72
 49 in logarithmic form.
7Solution: log 49 2

log5 1 0Solution:
2. Write 50
 1 in logarithmic form.

3. Write 10
 2

1
100
Solution: log10
1
100
  2

Solution: log16 4 
1
2
4. Finally, write 16
1
2
 4

Example 1:
Write log3 81 4 in exp onential form
Solution: 34
 81

Example 2:
Write log2
1
8
  3 in exp onential form.
Solution: 2
 3

1
8

Okay, now you try these next
three.
1. Write log10 100  2 in exp onential form.
3. Write log27 3 
1
3
in exp onential form.
2. Write log5
1
125

1. Write log10 100  2 in exp onential form.
Solution: 102
100

2. Write log5
1
125
Solution: 3 1
5
125



3. Write log27 3 
1
3
in exp onential form.
Solution: 27
1
3
 3

Solution:
Let’s rewrite the problem
in exponential form.
62
 x
We’re finished !
6Solve for x: log 2x 
Example 1

Solution:
5
y

1
25
Rewrite the problem in
exponential form.
Since
1
25
 5 2





5y
 5 2
y  2
5
1
Solve for y: log
25
y
Example 2

Example 3
Evaluate log3 27.
Try setting this up like this:
Solution:
log3 27  y Now rewrite in exponential form.
3y
 27
3y
 33
y  3

These next two problems
tend to be some of the
trickiest to evaluate.
Actually, they are merely
identities and
the use of our simple
rule
will show this.

Example 4
Evaluate: log7 72
Solution:
Now take it out of the logarithmic form
and write it in exponential form.
log7 72
 y First, we write the problem with a variable.
7y
 72
y  2

Example 5
Evaluate: 4log 4 16
Solution:
4
log 4 16
 y First, we write the problem with a variable.
log4 y  log4 16 Now take it out of the exponential form
and write it in logarithmic form.
Just like 23
 8 converts to log2 8  3
y  16

Example 1
Solve: log3 (4x 10)  log3 (x 1)
Solution:
Since the bases are both ‘3’ we simply set
the arguments equal.
4x10 x1
3x101
3x   9
x   3

Example 2
Solve: log8 (x2
14)  log8 (5x)
Solution:
Since the bases are both ‘8’ we simply set the arguments equal.
x2
14  5x
x2
 5x 14  0
(x  7)(x  2) 0
Factor
(x  7) 0 or (x  2) 0
x  7 or x  2 continued on the next page

Logarithmic function, equation and inequality

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