SOLUTION: 5 raised to log25(4X)=8

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Question 387862: 5 raised to log25(4X)=8
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
5%5Elog%2825%2C+%284x%29%29+=+8
If the logarithm was a base 5 logarithm the expression on the left would simplify easily. This is true because of what logarithms are. Logarithms are exponents, Base 5 logarithms are exponents for a 5. log%285%2C+%28q%29%29 is the exponent for 5 that results in q. So if we had an expression:
5%5Elog%285%2C+%284x%29%29
it would simplify to 4x because log%285%2C+%284x%29%29 is the exponent for 5 that results in 4x and we find log%285%2C+%284x%29%29 as the exponent for a 5!

So we are going to start by using the base conversion formula, log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29, to convert the base 25 logarithm into an expression of base 5 logarithms:
5%5E%28log%285%2C+%284x%29%29%29%2Flog%285%2C+%2825%29%29%29+=+8
Since log%285%2C+%2825%29%29 represents the exponent for 5 that results in 25 and since we know that the exponent for 5 that results in 25 is 3, this simplifies to:
5%5E%28log%285%2C+%284x%29%29%29%2F2%29+=+8
We are now close the the easily simplified 5 raised to the base 5 logarithm power we have been trying to get. The 2 needs to go. Here's how. Dividing by two is the same as multiplying by 1/2:
5%5E%28%281%2F2%29%2Alog%285%2C+%284x%29%29%29%29+=+8
Now we can use a property of logarithms, q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29, to move the 1/2 into the argument as an exponent:
5%5Elog%285%2C+%28%284x%29%5E%281%2F2%29%29%29+=+8
The left side is now in the easily simplified form. Since log%285%2C+%28%284x%29%5E%281%2F2%29%29%29 represents the exponent for 5 that results in %284x%29%5E%281%2F2%29 and since log%285%2C+%28%284x%29%5E%281%2F2%29%29%29 is the exponent on a 5, the left side must be %284x%29%5E%281%2F2%29. This gives us:
%284x%29%5E%281%2F2%29+=+8
We can now solve for x. Square both sides:
4x = 64
Divide by 4:
x = 16.