Equation problem, solution uses logarithm

[Loki123]

So this problem is very confusing to me. The first two pictures show how I tried to solve it, but even for that I used the solution as a guide. I would have never gotten the idea to use t like they did. In the end I also brought in logarithms, because they did (3rd picture), however, I still did not get the answer they did and it makes no sense to me. I feel like my solution with logarithms is a lot simpler... How did they get what they got? Is there another way to solve this? I don't think If I got this problem on an exam I would get the idea to use t like they did...

 

[Dr.Peterson]

So this problem is very confusing to me. The first two pictures show how I tried to solve it, but even for that I used the solution as a guide. I would have never gotten the idea to use t like they did. In the end I also brought in logarithms, because they did (3rd picture), however, I still did not get the answer they did and it makes no sense to me. I feel like my solution with logarithms is a lot simpler... How did they get what they got? Is there another way to solve this? I don't think If I got this problem on an exam I would get the idea to use t like they did...View attachment 29592View attachment 29593View attachment 29594

First, your answer is not wrong; it's equivalent to theirs. You can show that by using the change-of-base formula.

We generally avoid using fractional bases, for several reasons, so it's better to take the natural or common log of both sides rather than do what you did, but you are not wrong in what you did. It's just an uglier form.

Second, let's think about how they came up with their idea of t.



When I see this, my first observation is the x is in too many places, so I'd like to consolidate them somehow.

My second observation is that x is in the exponent with three different bases, so I can't yet use a substitution, as I expect to. But I see that the bases involve 2's and 3's, so they are related, and I wonder if I can get them to use the same base, which would most naturally be 3/2.

I can do that if I divide [imath]3^{x+1}[/imath] by [imath]2^{x+1}[/imath]. That suggests dividing the numerator and denominator by that, which works out nicely. If that hadn't worked, I would have tried something else!

See if these ideas help you out.

 

[Loki123]

First, your answer is not wrong; it's equivalent to theirs. You can show that by using the change-of-base formula.

We generally avoid using fractional bases, for several reasons, so it's better to take the natural or common log of both sides rather than do what you did, but you are not wrong in what you did. It's just an uglier form.

Second, let's think about how they came up with their idea of t.

View attachment 29595

When I see this, my first observation is the x is in too many places, so I'd like to consolidate them somehow.

My second observation is that x is in the exponent with three different bases, so I can't yet use a substitution, as I expect to. But I see that the bases involve 2's and 3's, so they are related, and I wonder if I can get them to use the same base, which would most naturally be 3/2.

I can do that if I divide [imath]3^{x+1}[/imath] by [imath]2^{x+1}[/imath]. That suggests dividing the numerator and denominator by that, which works out nicely. If that hadn't worked, I would have tried something else!

See if these ideas help you out.

yes, thank you. I solved it and now I understand.