solving logarithmic equations
TRANSCRIPT
LOGARITHMSSOLVING LOGARITHMIC EQUATIONS EXAMPLES
Updated: 2016/12/08File: 20161125-123944-rs2.2B-SolvingLogarithmicEqnExamples
Sigtunaskolan Humanistiska LäroverketSSHL
EXAMPLE: 20161124-152528
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Since the logarithms are of different bases,
log 2 x2+ log 4
√x � 9
we will need to modify one of them such that they both havethe same base.
Reason: This will be required so that we can use the LogarithmProduct of Common Base Powers Identity to solve the equation.
logb
m + logb
n � logb(mn)
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Since the logarithms are of different bases,
log 2 x2+ log 4
√x � 9
we will need to modify one of them such that they both havethe same base.
Reason: This will be required so that we can use the LogarithmProduct of Common Base Powers Identity to solve the equation.
logb
m + logb
n � logb(mn)
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Since the logarithms are of different bases,
log 2 x2+ log 4
√x � 9
we will need to modify one of them such that they both havethe same base.
Reason: This will be required so that we can use the LogarithmProduct of Common Base Powers Identity to solve the equation.
logb
m + logb
n � logb(mn)
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x︸¨̈︷︷¨̈︸z
� 9 for x.
Let’s change the base of log4√
2 to base 2.
z � log4√
x
4z�√
x
log2[4z]
� log2
[√x]
z log2 (4) � log2
(√x)
z(2) � log2
(√x)
z �12 log2
√x
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29
[x9/4
]4/9�
[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16
Example: 20161124-152528
Solve the equation log2 x2 + log4√
x � 9 for x.
Replacing log4√
x with 12 log2
√x
log2 x2+
[12
log2√
x]� 9
log2 x2+ log2
(√x)1/2
� 9
log2 x2+ log2
(x1/2
)1/2� 9
log2 x2+ log2
(x1/4
)� 9
log2
(x2 · x1/4
)� 9
log2 x9/4� 9
2[log2 x9/4
]� 2[9]
x9/4� 29[
x9/4]4/9
�[29]4/9
x �[29]4/9
x � 16