SOLUTION: Solve 100000^(4logx^5)

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Question 628160: Solve 100000^(4logx^5)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
100000%5E%284log%28%28x%5E5%29%29%29
First, what you posted is an expression. Expressions are simplified, not solved.

To simplify this we are going to transform the expression into the form
a%5Elog%28a%2C+%28q%29%29
We are going to do this because a%5Elog%28a%2C+%28q%29%29+=+q no matter what "a" and "q" are.

Since the base of "log" is 10 we want to express 100000 as a power of 10. Since 100000+=+10%5E5 this is fairly easy:
%2810%5E5%29%5E%284log%28%28x%5E5%29%29%29
The rule for exponents for a power of a power like this is to multiply the exponents:
10%5E%285%2A4log%28%28x%5E5%29%29%29
which simplifies to:
10%5E%2820log%28%28x%5E5%29%29%29
We have almost reached the desired form of a%5Elog%28a%2C+%28q%29%29. We don't wnat the 20. Fortunately there is a property of logarithms, q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29, which allows us to move a coefficient into the argument as its exponent:
10%5Elog%28%28%28x%5E5%29%5E20%29%29%29
Again we use the power of a power rule:
10%5Elog%28%28x%5E100%29%29
We now have the desired form, which simplifies to:
x%5E100