SOLUTION: log_x 100 = 1/2 log_x1024 = 10

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Question 238089: log_x 100 = 1/2
log_x1024 = 10
Answer by jsmallt9(2063) About Me  (Show Source):
You can put this solution on YOUR website!
For problems like yours, where the variable is the base or in the argument of the logarithm, the key is to rewrite the logarithmic equation in exponential form. To do this we need to know that log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq.

log%28x%2C+%28100%29%29+=+1%2F2
Rewriting this in exponential form we get:
100+=+x%5E%281%2F2%29
The equation is now solvable. We just square both sides:
%28100%29%5E2+=+%28x%5E%281%2F2%29%29%5E2
which simplifies to
10000+=+x

log%28x%2C+%281024%29%29+=+10
Rewriting this in exponential form we get:
1024+=+x%5E10
If you are a computer geek you may recognize that 2%5E10+=+1024. Otherwise we find the 10th root of each side:
root%2810%2C+1024%29+=+root%2810%2C+x%5E10%29
The left side is the same as 1024%5E%281%2F2%29 which we can find using a calculator. The right side simplifies to abs%28x%29. So now we have:
2+=+abs%28x%29
Solving this we get
x+=+2 or x+=+-2
And, since x is the base of a logarithm, we must reject x = -2. So the only solution is x = 2.