Question 393466: hello the question is
1/125=25^2x-1
FIND X:
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Assuming the equation is:
(If this is correct, then please put parentheses around multiple term exponents like 2x-1.)
When the variable in an equation is in an exponent, logarithms are often used to solve the problem. (And I will solve this way.) But if you can make both sides of the equation powers of the same number, then there is a much easier way to solve the equation. Since 125 and 25 are both powers of 5, this easier solution is possible for this equation. I'll start with the quick solution.
First we rewrite each side as powers of 5. Since  and since negative exponents are reciprocals,  . So now we have:
On the right side, the rule for raising a power to a power is to multiply the exponents. This gibes us:
which simplifies to:
The only way for two powers of 5 to be equal is for the exponents to be equal. So:
-3 = 4x-2
Adding 2 we get:
-1 = 4x
Dividing by 4 we get:
If we were not able to rewrite both sides of the equation as powers of the same number (or if we didn't notice that we could) then we use logarithms to solve equations like this. The base of logarithm is not really important. But if you want a decimal approximation for the answer, choose a base of logarithm that your calculator "knows", like base 10 or base e (aka ln). I'll use base 10 logarithms:
Next we use a property of logarithms,  , to move the exponent out in front as a coefficient. It is this very property that is the reason for using logarithms on equations like this. It allows us to move the exponent (where the variable is) out where we can solve for the variable. Using this property on the right side we get:
We can now solve for x. Dividing both sides by log(25) we get:
adding 1:
Dividing by 2:
Now we get out our calculators ans simplify the left side. You should get -0.25 (or some decimal extremely close to this). Since -0.25 = -1/4 this is the same answer we got the quick way.
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