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# If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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vaivish1723 wrote:
5^x – 5^(x- 3) = (124)(5^y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Oa is

Finding one variable in terms of another is very simple. Assume a value for one and find the other. Then look for the option that works.

Say, x = 3 (so that x-3 becomes 0 and we are left with simple calculations)

$$5^3 - 5^{3- 3} = (124)(5^y)$$
$$1 = 5^y$$
y = 0

Now put x = 3 in the options and find out which option gives you y = 0. Only C does so answer is (C)
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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alltimeacheiver wrote:
If $$5^x – 5^{(x- 3)} = 124*5^y$$, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

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

-----------------------------------------------------
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]
------------------------------------------------------------

Cheers,
Brent
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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alltimeacheiver wrote:
If $$5^x – 5^{(x- 3)} = 124*5^y$$, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Another option is to PLUG IN a value for x and see what kind of relationship we get between x and y.
There are two "nice" x-values to plug in. They are x = 0 and x = 3, since we can easily use these to evaluate 5^x and 5^(x-3). Of these two values, x = 3 is the easier one to plug in.
So, let's plug in x = 3
We get: 5^3 - 5^(3-3) = (124)(5^y)
Simplify to get: 125 - 1 = (124)(5^y)
Simplify to get: 124 = (124)(5^y)
Divide both sides by 124 to get: 1 = 5^y
Solve for y to get: y = 0

So, when x = 3, y = 0.

Now we'll check the answer choices to see which one satisfies this relationship.
A) y = x... So, we get 0 = 3 (NOPE)
B) y = x - 6... So, we get 0 = 3 - 6 (NOPE)
C) y = x - 3... So, we get 0 = 3 - 3 IT WORKS!
D) y = 2x + 3... So, we get 0 = 2(3) + 3 (NOPE)
E) y = 2x + 6... So, we get 0 = 2(3) + 6 (NOPE)

Cheers,
Brent
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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u can also plug in x=3. makes it very simple.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Hi Bunuel,
I', trying to understand your factorization, but I find some troubles when I have to factorize some value with exponent an expression like in the question (x-3 or x+4),
Could you explain me that or do you have a deck where I could dive into this argument?
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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mario1987 wrote:
Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Hi Bunuel,
I', trying to understand your factorization, but I find some troubles when I have to factorize some value with exponent an expression like in the question (x-3 or x+4),
Could you explain me that or do you have a deck where I could dive into this argument?

$$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3+3}-5^{x-3}=124*5^y$$ --> $$5^{x}-5^{x-3}=124*5^y$$.

Theory: math-number-theory-88376.html

Practice:
tough-and-tricky-exponents-and-roots-questions-125956.html
tough-and-tricky-exponents-and-roots-questions-125967.html

Hope it helps.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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$${5^{x}}\frac{124}{125}=124(5^y)$$
$$5^{x}=5^{y+3}$$
$$x=y+3-->y=x-3$$

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Given that 5^x – 5^(x- 3) = 124*5^y
So, 5^(x-3+3) - 5^(x- 3) = 124*5^y
So, 5^(x-3) [5^3 - 1] = 124*5^y
So, 5^(x-3)*124 = 124*5^y
So, 5^(x-3) = 5^y
So, x-3 = y

Hence option (C).

--
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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ok this is how you solve this

you have 5^x -5^x-3 = (124) (5^y)
step 1: 5^x (1-5^-3) =(124) (5^y)
Step 2: 5^x(1 - 1/5^3) = (124) (5^y)
Step 3: 5^x(1- 1/125) = (124) (5^y)
Step 4: 5^x (125-1)/125 = (124) (5^y)
Step 5: 5^x (124/125) = (124) (5^y)
Step 6: 5^x =[(124) (5^y) (125) ]/124
Step 7: 5^x = (125) (5^y)
Step 8: 5^x = 5^3 (5^y)
Step 9: x = 3 +y --> y = x -3
for explaining purposes I broke it down in to 9 steps. It should not take you more than 5 steps to solve this in term of y.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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peterpark wrote:
If 5^x - 5^x-3 = (124)(5^y), what is y in terms of x?

A) x

B) x - 6

C) x - 3

D) 2x + 3

E) 2x + 6

$$5^x - 5^{x-3} = (124)(5^y)$$

i.e. $$5^{x-3}(5^3 - 1) = (124)(5^y)$$

i.e. $$5^{x-3}(124) = (124)(5^y)$$

i.e. $$5^{x-3} = (5^y)$$

i.e. y = x-3

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
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alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

We can simplify the given equation:

5^x – 5^(x- 3) = 124*5^y

5^x – (5^x)(5^- 3) = 124*5^y

5^x(1 – (5^- 3)) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(124/125) = 124*5^y

5^x = 124*5^y*(125/124)

5^x = 5^y(125)

5^x = 5^y(5^3)

5^x = 5^y+3

x = y + 3

x - 3 = y

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Re: What is y in terms of x ? [#permalink]
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FillFM wrote:
If $$5^x$$ – 5^(x–3) = (124)$$(5^y)$$, what is y in terms of x ?

A) x

B) x – 6

C) x – 3

D) 2x + 3

E) 2x + 6

$$5^x$$ – 5^(x–3) = (124)$$(5^y)$$
or $$5^x$$ – $$\frac{5^x}{5^3}$$ = $$(125-1)$$$$(5^y)$$
or $$\frac{(5^x*5^3-5^x)}{5^3}$$ = $$(5^3-1)$$$$(5^y)$$
or $$5^x*(5^3-1)$$ = $$(5^3-1)$$$$(5^y)(5^3)$$
or $$5^x$$ = 5^(y+3)
==> $$x=y+3$$
==>$$y=x-3$$ ..................Thus the answer is option C.
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Re: What is y in terms of x ? [#permalink]
FillFM wrote:
If $$5^x$$ – 5^(x–3) = (124)$$(5^y)$$, what is y in terms of x ?

A) x

B) x – 6

C) x – 3

D) 2x + 3

E) 2x + 6

5^x - 5^(x - 3) = (124)(5^y)

We see that 5^(x - 3) is common to both terms on the left side of the equation, so we factor it out:

5^(x - 3)(5^3 - 1) = (124)(5^y)

5^(x - 3)(124) = (124)(5^y)

5^(x - 3) = 5^y

Recall that when the bases are equal, we can equate the exponents:

x - 3 = y

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
ScottTargetTestPrep wrote:
alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

We can simplify the given equation:

5^x – 5^(x- 3) = 124*5^y

5^x – (5^x)(5^- 3) = 124*5^y

5^x(1 – (5^- 3)) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(124/125) = 124*5^y

5^x = 124*5^y*(125/124)

5^x = 5^y(125)

5^x = 5^y(5^3)

5^x = 5^y+3

x = y + 3

x - 3 = y

Could you help me understand the specific steps you took to get from the first bold line to the second? I'm not sure how you managed to get rid of the fraction here. Thanks!
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
travelwithbrandon wrote:
ScottTargetTestPrep wrote:
alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

We can simplify the given equation:

5^x – 5^(x- 3) = 124*5^y

5^x – (5^x)(5^- 3) = 124*5^y

5^x(1 – (5^- 3)) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(124/125) = 124*5^y

5^x = 124*5^y*(125/124)

5^x = 5^y(125)

5^x = 5^y(5^3)

5^x = 5^y+3

x = y + 3

x - 3 = y

Could you help me understand the specific steps you took to get from the first bold line to the second? I'm not sure how you managed to get rid of the fraction here. Thanks!

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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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
travelwithbrandon
It's mentioned in post that it is a sub-600 level question however I find it better than sub 600 level.
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If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x? [#permalink]
Genoa2000 wrote:
What level is this question? Where can I see the level of the questions?

Not always I find it...

Its mentioned in the left hand corner just before the question starts. sub 600
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