Problem Solving

If 2^x = 5 and 4^y = 20, what is the value of x in terms of y?

A) y - 2
B) (y + 1)/2
C) 2y - 2
D) y/2 - 2
E) 2y + 2

Answer: C
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

Brent@GMATPrepNow wrote:If 2^x = 5 and 4^y = 20, what is the value of x in terms of y?

A) y - 2
B) (y + 1)/2
C) 2y - 2
D) y/2 - 2
E) 2y + 2

There are several approaches we can take. Here's the algebraic approach:
Given: 2^x = 5 and 4^y = 20

Take second equation and rewrite 4 as 2² to get: (2²)^y = 20
Simplify to get: 2^(2y) = 20

We now have:
2^(2y) = 20
2^x = 5

This means we can write: 2^(2y)/2^x = 20/5
Simplify: 2^(2y - x) = 4
Rewrite as: 2^(2y - x) = 2^2
So, it must be the case that 2y - x = 2
Add x to both sides: 2y = x + 2
Subtract 2 from both sides: 2y - 2 = x

Answer: C

Cheers,
Brent

Brent@GMATPrepNow wrote:If 2^x = 5 and 4^y = 20, what is the value of x in terms of y?

A) y - 2
B) (y + 1)/2
C) 2y - 2
D) y/2 - 2
E) 2y + 2

Answer: C
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

Here's a totally different approach:

GIVEN: 2^x = 5
2^2 = 4 and 2^3 = 8
Since 5 is BETWEEN 4 and 8, we know that x is BETWEEN 2 and 3.
From here, we can estimate.
Since 5 is closer to 4 than it is to 8, we know that x will be closer to 2 than it is to 3.
Let's say that x ≈ 2.3

GIVEN: 4^y = 20
4^2 = 16 and 4^3 = 64
Since 20 is BETWEEN 16 and 64, we know that y is BETWEEN 2 and 3.
From here, we can estimate.
Since 20 is closer to 16 than it is to 64, we know that y will be closer to 2 than it is to 3.
Let's say that y ≈ 2.1

ASIDE: As we'll see, it doesn't matter if our estimates are a little off

Now that we know that x ≈ 2.3 and y ≈ 2.1, we check the answers to see which one works.
That is, when we replace y with 2.1, which one yields an x-value that's close to 2.3

A) y - 2 = 2.1 - 2 = 0.1
This suggests that, when y = 2.1, x = 0.1. We want x = 2.3. ELIMINATE A.

B) y - 1/2 = 2.1 - 0.5 = 1.6
This suggests that, when y = 2.1, x = 1.6. We want x = 2.3. ELIMINATE B.

C) 2y - 2 = 2(2.1) - 2 = 4.2 - 2.1 = 2.1
This is VERY CLOSE to 2.3. KEEP C.

D) y/2 - 2 = 2.1/2 - 2 = some negative number
We want x = 2.3. ELIMINATE D.

E) 2y - 1/2 = 2(2.1) - 0.5 = 3.7
This suggests that, when y = 2.1, x = 3.7. We want x = 2.3. ELIMINATE E

By the process of elimination, the correct answer must be C

Cheers,
Brent