Question 636490: Solve 25^(n+1)+100=5^(2n-1)
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
This is a challenging problem. I don't have the time to go into extensive detail. I hope that the level of difficulty is a sign of your teacher's regard for your ability and that you will be able to follow what I'm doing despite the minimal comments.
Because I understand well what the rules of exponents are and how they can be used, I can see that this equation can be rewritten in terms of  . I am going to use the rules for exponents in the direction opposite to what you are used to seeing to do this:
If you have trouble seeing this, think about using the rules for exponents to go from the equation above backwards to the original equation. You should be able to see that in that direction we are using the rules in a "normal" way.
Next I am going to some simplifying and some simple manipulation:
Next I'm going to use a rule for exponents, backwards again, on  :
which simplifies to:
As you can see, we now have the equation in terms of
We can eliminate the fraction by multiplying both sides by 5:
Next I'll solve for . Subtracting from each side:
Exactly like 125k - k = 124k,  :
(If not now, then soon we should be seeing something that's not right.) Subtracting 500 we get:
Dividing by 124:
Do you see what's wrong? This equation says that a power of 25 equals a negative number. A power of 25, no matter what the exponent may be, can never be negative!! (Remember, negative exponents just mean reciprocals, not that something becomes negative.) So there is no way to make this equation true. There is no solution to this equation or the one we started with.
P.S. If we had an equation that said a power of 25 was a positive number, then we would use logarithms to solve that equation for n.
P.P.S. In response to the comment in your thank-you note...
There is no mistake in the solution above, Either "Microsoft Maths" is wrong about the answer being 2 or there is an error in the equation you posted. Here's a check to show you that 2 is not the solution:
Checking n = 2:
 The check fails. 2 in not a solution to this equation,
Again, either "Microsoft Maths" is wrong or there is an error in the equation you posted. I suspect that you posted the wrong equation because you don't often get problems with no solution and "Microsoft Maths" is probably not wrong.
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