SOLUTION: log 7 n=-1/2

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Question 398989: log 7 n=-1/2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Is the problem
log%28%287n%29%29+=+-1%2F2
or
log%287%2C+%28n%29%29+=+-1%2F2?
When posting logarithmic equations, please indicate the base of the logarithm more clearly. For example,
log%28%287n%29%29+=+-1%2F2
could be posted as any of the following:
common log of 7n = -1/2
base 10 log of 7n = -1/2
log base 10 of 7n = -1/2
or
log((7n)) = -1/2
log%287%2C+%28n%29%29+=+-1%2F2
could be posted as any of the following:
base 7 log of n = -1/2
log base 7 of n = -1/2
or
log(7, (n)) = -1/2
(If you put three left braces, "{" in front and three right braces, "}" behind the last ones, your equation would look like it should:
log%28%287n%29%29+=+-1%2F2
or
log%287%2C+%28n%29%29+=+-1%2F2
Tutors are more likely to help when a problem is clearly stated.

I'm going to guess that the correct equation is:
log%287%2C+%28n%29%29+=+-1%2F2
Solving an equation like this starts with rewriting the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this pattern on your equation we get:
n+=+7%5E%28-1%2F2%29
Now we just have to simplify the right side. If you have trouble with negative and/or fractional exponents, I find it helpful to factor the exponent in a certain way:
  1. If the exponent is negative, factor out a -1.
  2. If the exponent is fractional and the numerator is not a 1, then factor out the numerator.

Once factored, each factor tells us an operation to perform. Let's see how this works on:
n+=+7%5E%28-1%2F2%29
The exponent is negative so we will factor out a -1:
n+=+7%5E%28%28-1%29%2A%281%2F2%29%29
The exponent is fractional. But the numerator is a 1 so we are finished factoring. Looking at the factors of the exponent:
  • The -1 as an exponent tells us we will find a reciprocal.
  • The 1/2 as an exponent tells us that we will be finding a square root.

And since multiplication is Commutative we can do these operations in any order we choose! So let's see if we can find an "easy" order. The reciprocal of 7 is 1/7. So a reciprocal will mean we get a fraction. Since fractions are usually harder to work with than whole numbers we will save the reciprocal for last and do the square root first:
n+=+7%5E%28%281%2F2%29%2A%28-1%29%29
n+=+%287%5E%281%2F2%29%29%5E%28-1%29%29
n+=+%28sqrt%287%29%29%5E%28-1%29%29
n+=+1%2Fsqrt%287%29
As you may recall, final answers should not have a square root in a denominator. So we should rationalize the denominator. This can be done by multiplying the numerator and denominator by sqrt%287%29:
n+=+%281%2Fsqrt%287%29%29%28sqrt%287%29%2Fsqrt%287%29%29
which simplifies to:
n+=+sqrt%287%29%2F7

When solving logarithmic equations you must check your answers. You must ensure that your solution(s) make the bases and arguments of all logarithms positive. Any "solution" that makes a base or an argument of a logarithm zero or negative must be rejected. These rejected "solutions" can occur even if no mistakes have been made! This is why you must check.

Always use the original equation to check:
log%287%2C+%28n%29%29+=+-1%2F2
Checking n+=+sqrt%287%29%2F7:
log%287%2C+%28sqrt%287%29%2F7%29%29+=+-1%2F2
We can already see that the base and the argument are positive. So there is no reason to reject this solution. This is the required part of the check. The rest of the check will tell us if we have made mistakes. You are welcome to finish the check.

So your solution is n+=+sqrt%287%29%2F7.