SOLUTION: Please help me solve log base square root of 7 of 49=x

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Question 245869: Please help me solve log base square root of 7 of 49=x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28sqrt%287%29%2C+%2849%29%29+=+x
If you had an extremely amazing calculator which can figure out base sqrt%287%29 logarithms, you could just use your calculator to find x. But I doubt such a calculator exists so we will have to use some Math.

The key to the "easy" solution to this problem is to recognize that both sqrt%287%29 and 49 are powers of 7! If we see this then we can solve this by rewriting the equation in exponential form and then rewriting both sides as powers of 7 and then solving:
To rewrite the logarithmic equation in exponential form we have to know that log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this on our equation we get:
49+=+%28sqrt%287%29%29%5Ex
Now we'll rewrite this as powers of 7:
7%5E2+=+%287%5E%281%2F2%29%29%5Ex
Using the rule for exponents, %28a%5Ep%29%5Eq+=+a%5E%28pq%29, on the right side we get (since 1/2*x = x/2):
7%5E2+=+7%5E%28x%2F2%29
The only way this can be true is if the exponents are equal:
2+=+x%2F2
Multiplying both sides by two we get:
4+=+x

If we don't recognize that we have powers of 7 to work with, we can still solve the problem. Again we start by rewriting the equation in exponential form (so we can get rid of the logarithm we cannot use a calculator to find):
49+=+%28sqrt%287%29%29%5Ex
The variable is now in an exponent. The most common way to solve for a variable in an exponent is to use logarithms. Now we find the logarithm of each side using a base that our calculator can handle. Base 10 and/or base e (aka natural logarithms) are the logarithms most common to calculators (among those that can do logarithms at all). Base 10 logarithms are probably the most common so this is what we will use:
log%28%2849%29%29+=+log%28%28%28sqrt%287%29%29%5Ex%29%29
Next we use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent in the argument out in front. (This is how we get the variable out of the exponent.):
log%28%2849%29%29+=+x%2Alog%28%28sqrt%287%29%29%29
Now we divide by log%28%28sqrt%287%29%29%29:
log%28%2849%29%29%2Flog%28%28sqrt%287%29%29%29+=+x
and finally we get out our calculators:
log%28%2849%29%29%2Flog%28%282.6457513110645906%29%29+=+x
1.6901960800285137%2F0.4225490200071284+=+x
4.0000000000000002+=+x

Note that is a tiny bit different from the exact answer which is 4. This is what happens when you start using your calculator on square roots and logarithms. You only get decimal approximations (even with 16 decimal places) of the correct numbers and you end up with small errors like this. If you need exact answers we need to avoid caluclators. We would either need to figure out the first solution (which did not use a calculator) or use the second solution and stop before you use a calculator. In other words stop at: log%28%2849%29%29%2Flog%28%28sqrt%287%29%29%29+=+x

P.S. If we had used natural logarithms instead of base 10 we would get:
ln%2849%29%2Fln%28sqrt%287%29%29+=+x
which looks different but, if you use your calculator, still works out to be a number extremely close to 4!

P.S.S. Note that
log%28%2849%29%29%2Flog%28%28sqrt%287%29%29%29+=+x
and
ln%2849%29%2Fln%28sqrt%287%29%29+=+x
are simply the formulas for converting the base of a logarithm, the first one from base sqrt%287%29 to base 10 and the second one from base sqrt%287%29 to base e!