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Trigonometry
Logarithm Equation Examples
Logarithm Equation Example 1
Question:
If log
_{16}
25 = xlog
_{2}
5 then x equals ?
Solution
Let y = log
_{16}
25 = xlog
_{2}
5
when y = log
_{16}
25
then 16
^{y}
= 25
then (2
^{4}
)
^{y}
= 5
^{2}
then (2
^{y}
)
^{4}
= 5
^{2}
...equation (1)
when y = xlog
_{2}
5
then y = log
_{2}
5
^{x}
then 2
^{y}
= 5
^{x}
...equation (2)
Substitute the expression for 2
^{y}
in equation (2) into equation (1)
(5
^{x}
)
^{4}
= 5
^{2}
then 5
^{4x}
= 5
^{2}
then 4x = 2 then x = 1/2
Logarithm Equation Example 2
Question:
log
_{2}
16
^{-0.5}
= ?
Solution
log
_{2}
16
^{-0.5}
= - 0.5 log
_{2}
16 = - 0.5 log
_{2}
2
^{4}
= - 0.5 × 4 log
_{2}
2 = - 0.5 × 4 = - 2
formulas used
(1). log
_{a}
b
^{x}
= x log
_{a}
b
(2). log
_{a}
a = 1
Logarithm Equation Example 3
Question:
What is the domain of the function log(2x
^{2}
- 3) ?
Solution
The domain of a log(x) function is that the variable x must be greater than zero.
From this, we have 2x
^{2}
- 3 > 0
2x
^{2}
> 3
x
^{2}
> 1.5
then x < -1.22 or x > 1.22
Therefore, the domain of the given function is x must be larger than 1.22 or smaller than -1.22
Exponential Equation Example 1
Question:
Solving the equation x
^{lg x}
= 1000 x
^{2}
Solution:
From the definition of logarithm, if a
^{x}
= b (a > 0, and a is not 0) then x = log
_{a}
b
If a = 10, then log
_{10}
b, marked as lg b,
Take the lg of both side ( note: lg b is log
_{10}
b)
lg(x
^{lg x}
) = lg(1000 x
^{2}
)
(lg x)(lg x) = lg1000 + lg x
^{2}
(lg x)
^{2}
= lg 10
^{3}
+ 2 lg x
(lg x)
^{2}
- 2 lg x - 3 = 0
Factoring this quadratic equation, (lg x + 1)(lg x - 3) = 0
lg x + 1 = 0 or lg x - 3 = 0
For lg x + 1 = 0 => lg x = -1 => 10
^{-1}
= x => x = 1/10
For lg x - 3 = 0 => lg x = 3 => 10
^{3}
= x => x = 1000
formulas used:
log
_{a}
b
^{x}
= x log
_{a}
b
log
_{a}
a = 1
log
_{a}
(MN) = log
_{a}
M + log
_{a}
N
Exponential Equation Example 2
Question:
f(x) = 2
^{x}
+ 3 is a function. If f(x) = 10, find x = ?
Solution
Given f(x) = 10, then 2
^{x}
+ 3 = 10 => 2
^{x}
= 7
Take log to both sides
log2
^{x}
= log 7 => x log2 = log 7
x = log 7/log 2 = 0.8451/0.3010 = 2.8076