SOLUTION: 10=50(1/20^(1/13)(t) how do you get to the next step when there is a variable for an exponent with a number?

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Question 269804: 10=50(1/20^(1/13)(t)
how do you get to the next step when there is a variable for an exponent with a number?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Your equation makes no sense. The parentheses are not even balanced. So I cannot help you with it.

But here are some general tips:
  1. Isolate the base and its exponent with a variable.
  2. Find the logarithm of each side of the equation. Generally the base you use does not matter. If you want a decimal approximation of the answer, use a base for the logarithms which your calculator "knows" (like base 10 or base e (ln)). Sometimes it is helpful to choose the base of the logarithm to match the base on which there is an exponent with the variable.
    3. After steps #1 and #2, you should have an equation where one side is the log of a number with an exponent that has the variable for which you're trying to solve. On this logarithm, use the 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 you get a variable out of an exponent.
  3. Solve the resulting equation for the variable.

Here's an example:
8+=+3%2A4%5E%28x-3%29+-+7
1. Isolate the base and its exponent. Add 7 to both sides:
15+=+3%2A4%5E%28x-3%29
Divide both sides of the equation by 3:
5+=+4%5E%28x-3%29
2. Find the log of each side. If we want a simple, exact answer, use base 4 logarithms. If we want a decimal approximation, use base 10 or base e logarithms. I will base 4. (At the end I will use base 10 logarithms so you can see that, too.)
log%284%2C+%285%29%29+=+log%284%2C+%284%5E%28x-3%29%29%29
3. Use the property of logarithms to move the exponent out in front:
log%284%2C+%285%29%29+=+%28x-3%29log%284%2C+%284%29%29
Since log%284%2C+%284%29%29+=+1 (which is why I chose base 4 logarithms) this becomes:
log%284%2C+%285%29%29+=+x-3
4. Solve the equation. All we need to do is add 3 to each side:
log%284%2C+%285%29%29+%2B+3+=+x
This is an exact answer to the example problem.

At step #2, if we use base 10 logarithms instead, we get:
log%28%285%29%29+=+log%28%284%5E%28x-3%29%29%29
Then, using the property of logarithms we get:
log%28%285%29%29+=+%28x-3%29log%28%284%29%29
And to solve for x we divide both sides by log(4):
log%28%285%29%29%2Flog%28%284%29%29+=+x-3
and add 3 to each side:
log%28%285%29%29%2Flog%28%284%29%29+%2B+3+=+x
This is also an exact answer to the example equation. And if we use our calculators we can find a decimal approximation of the answer:
0.6989700043360188%2Flog%28%284%29%29+%2B+3+=+x
0.6989700043360188%2F0.6020599913279624+%2B+3+=+x
1.1609640474436812+%2B+3+=+x
4.1609640474436812+=+x