1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.
Problem 5: Solve for x in the equation
Answer: is the exact answer and x-0.7639320225 is an approximate answer.
Solution:
When the base is the same as the base of the logarithm, the above equation can be simplified to
There are two exact answer: and and there are two approximate answers:
However only one of the answers is valid.
One of the answers ( ) is out of our specified domain of the set of all real numbers such that (x
> -2).
Therefore, the exact and approximate answers are:
Check: Let's substitute the value in the original equation and determine whether the left side of the equation equals the right side of the equation after the substitution. In other words, does
or
or
or
or
The answer checks. We can also check our solution with the approximate answer. Does
or does
In another word,
Since the value of the left side of the original equation equals the right
side of the original equation when we substitute the exact and the
approximate value of x, We have proved the answer.
Let's illustrate why we had to throw out one of the answers. Let's check to
see if the approximate solution (the one we discarded) works:
At this point we have to stop because we cannot take the log of a negative number. We simply cannot calculate the value if any of the terms are undefined.
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