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If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or

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Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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05 Nov 2014, 10:17
1
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Difficulty:

85% (hard)

Question Stats:

57% (02:28) correct 43% (02:08) wrong based on 137 sessions

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If $$6^A = 2$$, $$6^B = 5$$, and $$6^Q = 15$$, then express Q in terms of A and/or B.

(A) 3B
(B) B + 3
(C) 5A + B
(D) $$B^2 - 2B$$
(E) B - A + 1

For the solution to this, as well as for a set of challenging problems on exponents, see:
https://magoosh.com/gmat/2014/challengi ... and-roots/

Mike

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Mike McGarry
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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11 Nov 2014, 01:43
6
1
$$6^q = 15 = \frac{15 * 2}{2} = \frac{30}{2} = \frac{5 * 6}{2}$$ .................. (1)

$$6^A = 2, 6^B = 5$$>> Placing these values in (1)

$$6^q = \frac{6^b * 6}{6^a}$$

$$6^q = 6^{(b+1-a)}$$

Bases are same; equating the powers

q = b+1-a

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Posts: 275
Re: If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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06 Nov 2014, 00:29
3
1
mikemcgarry wrote:
If $$6^A = 2$$, $$6^B = 5$$, and $$6^Q = 15$$, then express Q in terms of A and/or B.
(A) 3B
(B) B + 3
(C) 5A + B
(D) $$B^2 - 2B$$
(E) B - A + 1

For the solution to this, as well as for a set of challenging problems on exponents, see:
https://magoosh.com/gmat/2014/challengi ... and-roots/

Mike

6^A=2 ----------------------1)
6^B=5 -----------------------2)
6^Q= 15 ---------------------3)

multiplying 1,2,3 together, we have
6^(A+B+Q)= 2*5*15
= 2*3*5^2
=6*5^2
from 2, we have 5^2= 6^2B

=6*6^2B
=6^2B+1
thus
A+B+Q=2B+1
Q=B-A+1
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Posts: 7206
Re: If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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06 Nov 2017, 10:06
1
mikemcgarry wrote:
If $$6^A = 2$$, $$6^B = 5$$, and $$6^Q = 15$$, then express Q in terms of A and/or B.
(A) 3B
(B) B + 3
(C) 5A + B
(D) $$B^2 - 2B$$
(E) B - A + 1

For the solution to this, as well as for a set of challenging problems on exponents, see:
https://magoosh.com/gmat/2014/challengi ... and-roots/

Mike

Hi
see if all can be brought to base 6..
now we have 2,5,15 and these can be arranged as $$\frac{15}{5}*2=6$$
substitute corresponding values...
$$\frac{15}{5}*2=\frac{6^Q}{6^B}*6^A=6^{Q+A-B}=6^1........Q+A-B=1........Q=B+1-A$$
E
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Posts: 122
Re: If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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06 Nov 2017, 11:07
6^A=2, 6^B=5, and 6^Q=15
applying log base 6 (log6) to both sides of the equations
log6(6^A) = log6(2)
log6(6^B)=log6(5)
log6(6^Q)=log6(15)
using log properties

A = log6(2)
B = log6(5)
Q = log6(15) and just scrolling the answers down for anyone who knows logs first answer to check is E (as others lead to much greater values of Q)
log6(15) = log6(5) - log6(2) + 1
1 is log6(6) so we can rewrite
log6(15) = log6(5) - log6(2) + log6(6) -> applying another log property
log6(15) = log6 ((5/2)*6) = log6 (15)
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Joined: 15 Nov 2018
Posts: 15
If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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15 Nov 2018, 08:46
chetan2u wrote:
mikemcgarry wrote:
If $$6^A = 2$$, $$6^B = 5$$, and $$6^Q = 15$$, then express Q in terms of A and/or B.
(A) 3B
(B) B + 3
(C) 5A + B
(D) $$B^2 - 2B$$
(E) B - A + 1

For the solution to this, as well as for a set of challenging problems on exponents, see:

Mike

Hi
see if all can be brought to base 6..
now we have 2,5,15 and these can be arranged as $$\frac{15}{5}*2=6$$
substitute corresponding values...
$$\frac{15}{5}*2=\frac{6^Q}{6^B}*6^A=6^{Q+A-B}=6^1........Q+A-B=1........Q=B+1-A$$
E

Hi Chetan,

Can you let me know if this strategy is viable? I am not strong at math and rely on a lot of shortcuts to get to correct answers.

For this question I assumed:
A = 0.5
B = ~0.8
Q = ~1.5

Upon plugging these numbers in to the answer key, [E] was the only answer even CLOSE to correct.

Can you let me know if this is a strategy I can continue to employ?

Thanks
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If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or  [#permalink]

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15 Nov 2018, 09:14
mikemcgarry wrote:
If $$6^A = 2$$, $$6^B = 5$$, and $$6^Q = 15$$, then express Q in terms of A and/or B.

(A) 3B
(B) B + 3
(C) 5A + B
(D) $$B^2 - 2B$$
(E) B - A + 1

For the solution to this, as well as for a set of challenging problems on exponents, see:
https://magoosh.com/gmat/2014/challengi ... and-roots/

Mike

Used the options to get to the answer:

1) => $$6^(3B)$$
=> $$(6^B)*(6^B)*(6^B)$$
=> 5*5*5=125
So incorrect

2) $$(6^(B+3))$$
=> $$(6^B)*(6^3)$$
=> 5*216=1080
So incorrect

3) $$(6^(5A+B))$$
=> $$(6^5A)*(6^B)$$
=> $$(2^5)*5$$
=> 32*5
=> 160
So incorrect

4) (6^$$B^2 - 2B$$)
=> $$\frac{(6^(B^2))}{(6^2B)}$$
=> $$\frac{(5^B)}{25}$$
Cant be calculated with the given information

5) 6^(B-A+1)
=> $$\frac{((6^B)*6)}{(6^A)}$$
=> $$\frac{(5*6)}{2}$$
=> 15
So correct and the answer is E
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If 6^A = 2, 6^B = 5, and 6^Q = 15, then express Q in terms of A and/or &nbs [#permalink] 15 Nov 2018, 09:14
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