Question 448833: Well ive been stuck on a problem : log(x^2-5x)-logx=log2
the problem is i get x=LOG7 instead of it actually equaling just x=7 i need to know how to fully complete the problem correct
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
log(x^2-5x)-log(x)=log(2)
First, the difference of two logarithms is equivalent to the log of the first quantity divided by the second quantity. In this problem, the log(x^2-5x) is the first term. The log(x) is subtracted from this. This subtraction is equivalent to:
log[(x^2 - 5x)/x]
Performing the division within the [ ... ] brackets, that is dividing x into each of the two terms in the parentheses, results in:
log[x - 5]
This simplification of the left side of the problem converts the problem to:
log[x - 5] = log[2]
Look carefully at this statement of the problem. The log operator is equivalent on both sides. Therefore, for this equation to be true, the terms in the brackets on both sides must be equal. Basically what you have is:
log[A] = log[A]
which is true only if A is equal to A.
So what we have reduced the problem to is:
[x - 5] = [2]
Solve this equation by adding 5 to both sides to eliminate the -5 on the left side and get:
x = 2 + 5 = 7
You can check this by substituting 7 for x in the original problem as follows:
log(x^2-5x)-log(x)=log(2)
log[7^2 -(5*7)] - log(7) = log(2)
Do the math in the first set of brackets:
log[49 - 35] - log(7) = log(2)
49 - 35 = 14 so that the first term becomes:
log[14] - log[7] = log[2]
You can use a calculator to determine each of the logs:
1.146128036 - 0.84509804 = log(2)
The subtraction on the left side results in:
0.301029996 = log(2)
Now use your calculator to find log(2) and the equation is:
0.301029996 = 0.301029995
These are equal within calculator accuracy. This shows that x = 7 is correct.
Hope this helps you to understand the problem better.
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