Question

Can someone explain to me how to solve this problem?

I didn't think you could change the base values, but this is what this problem is implying.

Answer 1

I'm a bit unfamiliar with what is meant by \(\text{exp}_3\) and \(\text{exp}_5\). Could you briefly enlighten me? Does it mean the functions \(3^x\) and \(5^x\)?

Answer 2

Yes it does @KingGeorge

Answer 3

It has to do with logarithmic functions. My professor explained to me that log functions are the inverses of the exponential functions (which are the ones you're looking at).

An example he provided for me was this: exp[2](x) = 2^x. (with [2] being the base here), so it's pretty much the same thing as logs, just slightly different. At least, that's my understanding of it.

Answer 4

Alright. In that case, if we write the problem in a slightly more conventional notation, we're asked to find a function \(g\) such that\[3^x=5^{g(x)}.\]Since you seem to be working with logarithms, it might be a good idea to look at that. In particular, there's a property of logs that allow them to be "cancelled" with exponential functions. This property says that\[\Large \log_a(a^x)=x\quad\text{and}\quad a^{\log_a(x)}=x\]

Answer 5

So if you use this property, you could look at your original problem, and hopefully think that we need to either change the base, or completely get rid of the 5. This property of logs allows you to completely remove it. So to get you started, \(g(x)\) should probably be of the form\[\log_5(?)\]as this will cancel the 5, and you will only be left with the "?". What do you think should be where the question mark is?

Answer 6

Well, if we want the problem to equal 3^x, would ?= 3^x?

Something like this" 3^x=5^log5(3^x)

Would this be correct? Or am I totally off the track here?

Answer 7

That's exactly what I would put.

Before we call it at that though, we need to check one more thing. The questions specifically asks for a function \(g\) that has the real numbers as a domain. So we need to make sure that \(g\) is defined for every real number. Since we have a logarithm mixed in, that's not always necessarily true.

Fortunately, the first thing we do to a value \(x\), look at \(3^x\). Since this number is always positive for any real number \(x\), we have that \(\log_5(3^x)\) is always defined.

Answer 8

Ok, this is much clearer to me now, and I can see that this makes sense. So, at this point, I need to write the function in g(x) form, right?

Answer 9

Well the question only asks for the function \(g\), so in my mind it would be enough to say that \[g(x)=\log_5(3^x)\]is the function, with possibly a brief explanation of why it works.

Answer 10

Ok, so then I could demonstrate by using it how this will make the the problem true. Thank you so much, I finally understand what this means now! You've been really helpful :)

Answer 11

You're very welcome.