SOLUTION: I am having a lot of difficuties solving this because of the sqrt. Write in exponential form: log7 sqrt7 = 1/2

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Question 410313: I am having a lot of difficuties solving this because of the sqrt.
Write in exponential form:
log7 sqrt7 = 1/2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
In general log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Once you can look at these as patterns problems like this will become very easy.

When you look at log%28a%2C+%28p%29%29+=+q, try not to focus on the specific letters. The specific letters do not matter! The specific letters are just place-holders. In this case, the "a" is a place-holder for the base of the logarithm, the "p" is a place-holder for the argument of the logarithm, and the "q" is the place-holder for whatever the logarithm is equal to.

These place-holders can be replaced by any legitimate Mathematical expression! (I'll explain the "legitimate" part shortly.)

Once you get the idea that the letters are just place-holders, then you will understand that my use of the particular letters "a", "p" and "q" is irrelevant. I could have used any three letters! It is highly likely that your textbook uses different letters for these equations.

And not only can we use any letters in these three locations, we can use any (legitimate) expressions. The only thing that is important is that we follow the pattern described by:
log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq
The pattern here "says"
  • Whatever the base is in the logarithmic form, the "a", becomes the base with an exponent in the exponential form.
  • Whatever the logarithm is in the logarithmic form, the "q", becomes the exponent in the exponential form.
  • Whatever the argument of the logarithm is in logarithmic form, the "p", becomes what the exponential term is equal to in the exponential form.
And by "whatever" I mean exactly that. These expressions can be anything (that is legitimate)!

Let's see some examples:
log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq
log%28b%2C+%28k%29%29+=+z is equivalent to k+=+b%5Ez
log%28%28s-h%29%29+=+-2 is equivalent to s-h+=+10%5E%28-2%29 (because the implied base of log is 10)
ln%2812%29+=+3g%2B4 is equivalent to 12+=+e%5E%283g%2B4%29 (because the implied base of ln is the special number "e")
log%28v%2C+%28x-4%29%29+=+j is equivalent to x-4+=+v%5Ej
log%28d%2C+%28120%29%29+=+1%2F2 is equivalent to 120+=+d%5E%281%2F2%29
log%28y%2C+%28x%5E2-4x%2B2%29%29+=+-9 is equivalent to x%5E2-4x%2B2+=+y%5E%28-9%29
log%288x-4y%2Bz%2C+%2830%29%29+=+4 is equivalent to 30+=+%288x-4y%2Bz%29%5E4
log%28w%2C+%28log%284%2C+%28r%29%29%29%29+=+8.3 is equivalent to log%284%2C+%28r%29%29+=+w%5E8.3 which could then be written as r+=+4%5E%28%28w%5E8.3%29%29
and...
log%287%2C+%28sqrt%287%29%29%29+=+1%2F2 is equivalent to sqrt%287%29+=+7%5E%281%2F2%29

Note 1: This pattern also works in reverse. So we can use it to convert from exponential form into logarithmic form.

Note 2: "Legitimate" expressions. There are certain expressions that should never occur. Among these things that should never happen are:
  • Division by zero.
  • Bases of logarithms that are zero or one or anything negative,
  • Arguments of logarithms should never be zero or any negative number,
(Note: this is not a comlpete list!)

So when we replace the "a" in log%28a%2C+%28p%29%29+=+q we can replace it with anything except something that is equal to 0, 1 or any negative number. And when we replace the "p" in log%28a%2C+%28p%29%29+=+q, we can replace it with anything except something that is equal to 0 or any negative number.