5^x−5^(x−3)=124∗5^y
5^x−5^(x−3)=(125-1)∗5^y
5^x−5^(x−3)=(5^3-1)*5^y
5^x−5^(x−3)=5^(3+y)-5^y
x=3+y or x-3=y

C
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So what is the mistake in this method:

5^x - 5^(x - 3) = (5^3 - 5^0) * 5^y
5^x - 5^(x - 3) = (5^3 - 1) * 5^y
5^x - 5^(x - 3) = 5^3y - 5^y

3y = x AND y = x-3 - this is enough for correct answer
x = y+3
y + 3 = 3y
y = 1.5
x =4.5

Clearly, something is off with the above, but I have no idea what!

One option is to rewrite the left side of the equation by factoring out 5^(x-3)
So, we get: 5^(x-3)[5^3 - 1] = (124)(5^y)
Evaluate to get: 5^(x-3)[125 - 1] = (124)(5^y)
Simplify to get: 5^(x-3)[124] = (124)(5^y)
Divide both sides by 124 to get: 5^(x-3) = 5^y
So, x-3 = y

Answer: C

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ASIDE: A lot of students struggle to see how we can factor 5^x - 5^(x-3) to get 5^(x-3)[5^3 - 1]
Sure, they may be okay with straightforward factoring like these examples:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
But they have problems when the exponents are variables.

IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the smaller exponent.

So, in the expression 5^x - 5^(x-3), the term with the smaller exponent is 5^(x-3, so we can factor out 5^(x-3)

Likewise, w^x + x^(x+5) = w^x(1 + w^5)
And 2^x - 2^(x-2) = 2^(x-2)[2^2 - 1]
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Cheers,
Brent
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Hi,

I don't understand how all yall are factoring 5^x-3 out of 5^x-5^x-3 - isn't the only common factor 5^x???? I am getting 5^x(1-5^-3) for the left side of the equation

Response:

Notice that in the first bold line, the fraction 124/125 is not in the exponent; that fraction is simply being multiplied by 5^x (as a matter of fact, nothing besides x is in the exponent on and after the third line). Thus, to get rid of the fraction 124/125 on the left hand side, I simply multiplied each side of the equation by 125/124.
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