Problem Solving

How do you solve for X in this equation?
Do you have any sources where I can read more about it?

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

Hmm, here's how I approached it, but got stuck at the end

3^(x-1) = (3^x)/3

3^x - 3^(x-1)
= 3^x - (3^x/3) = 3^5

Multiply both sides by 3

=> 3^(x+1) - 3^x = 3^6

Now, 3^(x+1 ) = 3^x * 3

Factoring out 3^x

3^x(3 - 1) = 3^6
=> 3^x = 3^6 x 0.5

This gives a non-integer value for x

Could you please give the OA and the source of the question?

kmittal82 wrote:Hmm, here's how I approached it, but got stuck at the end

3^(x-1) = (3^x)/3

3^x - 3^(x-1)
= 3^x - (3^x/3) = 3^5

Multiply both sides by 3

=> 3^(x+1) - 3^x = 3^6

Now, 3^(x+1 ) = 3^x * 3

Factoring out 3^x

3^x(3 - 1) = 3^6
=> 3^x = 3^6 x 0.5

This gives a non-integer value for x

Could you please give the OA and the source of the question?

Thanks!

The source is the GMAT Prep exams.

Hi Andre!

Search the forum for information on solving exponential equations. Look for expert replies.

you can check also this link:
https://www.purplemath.com/modules/solvexpo.htm

let us calculate it:

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

3^x - (1/3)*3^x = 3^5

3^x(1 - 1/3) = 3^5

2/3 * 3^x = 3 ^5

2 * 3^x = 3^6

3^x = (3^6)/2

log3 (3^x) = log3 ((3^6)/2)

x = log3 (3^6) - log3 (2)

x = 6 - log3 (2)


what are the answer choices?

----
log3 (2) = log10 (2)/log10 (3)
x = 6 - log10 (2)/log10 (3)
x = 6 - 0.3/0.48
x = 6 - 0.625
x = 5.375

hope it helps!

Best,
Maciek

Maciek wrote:Hi Andre!

Search the forum for information on solving exponential equations. Look for expert replies.

you can check also this link:
https://www.purplemath.com/modules/solvexpo.htm

let us calculate it:

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

3^x - (1/3)*3^x = 3^5

3^x(1 - 1/3) = 3^5

2/3 * 3^x = 3 ^5

2 * 3^x = 3^6

3^x = (3^6)/2

log3 (3^x) = log3 ((3^6)/2)

x = log3 (3^6) - log3 (2)

x = 6 - log3 (2)


what are the answer choices?

----
log3 (2) = log10 (2)/log10 (3)
x = 6 - log10 (2)/log10 (3)
x = 6 - 0.3/0.48
x = 6 - 0.625
x = 5.375

hope it helps!

Best,
Maciek

But aren't logarithms outside the scope of the GMAT ?

you're right!

logarithms are outside the scope of the GMAT

what are the aswer choices?

I was thinking the same thing?!

Hey guys,

Interesting question - my hunch is that, to be a GMAT question, it would need one more element that may not have been copied over. You're right that logarithms are out of scope.

Check this out:

3^x - 3^(x-1) can be written as 3^(x-1) * 3 - 3^(x-1). I do that so that we can factor out 3^(x-1) - keep in mind that we're generally pretty terrible at adding/subtracting exponents, so when you're asked to do that you should try to factor so that you can do what we're good at - multiplying exponents!

If you do that and factor out 3^(x-1), you have:

[3^(x-1)] (3 - 1) = 3^5
[3^(x-1)] * 2 = 3^5

Well, we know that we can't do that with integer values, because the presence of 2 on the left means that the left hand side will be even, and the 3^5 on the right must be odd, so my hunch is that there's a 2 somewhere else in the original problem to allow this equation to balance with integers...


If it's written as given above, though, you'd see noninteger answer choices, and you could get away with estimating those. I'd do it this way - the 2 on the left hand side is going to make estimation pretty easy if we can combine the 3s on the right, so I'd divide both sides by the 3 term from the left:

2 = [3^5 / 3^(x-1)]

Then let's break that 3^(x-1) into 3^x * 3^(-1) so that we can get x on its own:

2 = 3^5 / (3^x * 3^-1)

2 = 3^6 / 3^x

Now you can estimate - if x were 5, then the right hand side would reduce to 3 (3^6 / 3^5 = 3). If it were 6, it would reduce to 1 (3^6 / 3^6). So we know that x is between 5 and 6, and that should give you enough to estimate the answer.



So...that's THAT problem, but let me leave it with a bigger takeaway:


When you're asked to add/subtract exponents, recognize that you're not very good at that. What you are good at is multiplying/dividing exponents, so look for opportunities to factor the addition/subtraction so that you can put the equation in multiply/divide form!

andre.heggli wrote:

kmittal82 wrote:Hmm, here's how I approached it, but got stuck at the end

3^(x-1) = (3^x)/3

3^x - 3^(x-1)
= 3^x - (3^x/3) = 3^5

Multiply both sides by 3

=> 3^(x+1) - 3^x = 3^6

Now, 3^(x+1 ) = 3^x * 3

Factoring out 3^x

3^x(3 - 1) = 3^6
=> 3^x = 3^6 x 0.5

This gives a non-integer value for x

Could you please give the OA and the source of the question?

Thanks!

The source is the GMAT Prep exams.

@ andre.heggli - Going forward, can you please write the source when you post the question? Some of us are saving the GMAT Prep Test for later and solving a question from the test gives away a potential question from the test. Thanks!