Exponential Problem -Problem solving help needed

[jd6199]

SOURCE: PEARSON SAMPLE TEST #1

2^x - 2^(x-2) = 3(2^13)

What is the value of x?

A. 9
B. 11
C. 13
D. 15 (correct answer)
E 17

Can someone show me the work of how to calculate this?

[g2000]

Let y = 2^x and we have

y - y/4 = 3 * (2^13)
3y / 4 = 3 * (2^13)
y = 2^13 *4
y = 2^15

plug it back to the original equ.

2^15 = 2^x
x = 15 (ans)

[docxyz]

Let y = 2^x and we have

y - y/4 = 3 * (2^13)
3y / 4 = 3 * (2^13)
y = 2^13 *4
y = 2^15

plug it back to the original equ.

2^15 = 2^x
x = 15 (ans)

how did u get y/4?.. kindly explain.

[Ian Stewart]

Let y = 2^x and we have

y - y/4 = 3 * (2^13)
3y / 4 = 3 * (2^13)
y = 2^13 *4
y = 2^15

plug it back to the original equ.

2^15 = 2^x
x = 15 (ans)

how did u get y/4?.. kindly explain.

Notice that:
2^(x-2) = (2^x)*(2^(-2)) = (2^x)*(1/4)

by the familiar exponent rules. If y = 2^x, then 2^(x-2) will equal y/4.

Still, I find it unnecessary to introduce a new letter y here. We know that:

2^x - 2^(x-2) = 3(2^13)

If you see that you can factor out a 2^(x-2) on the left side of this equation, the rest of the algebra becomes easy enough:

(2^(x-2))*(2^2 - 1) = 3(2^13)
(2^(x-2))*3 = 3(2^13)
2^(x-2) = 2^13
x-2 = 13
x = 15

[Abdulla]

How did you factor the left side to get this (2^(x-2))*(2^2 - 1) = 3(2^13)