Find the domain of the logarithmic function.

[Fufufuwie]

Hello!

Here is the problem I need a bit of help with.

4. Find the domain of the logarithmic function: \(\displaystyle \, f(x)\, =\, \log_4\, \left(\,\dfrac{x\, +\, 6}{x^2\, -\, 1}\,\right)\)

So I believe I am supposed to solve the problem and I got as far as having the following below:

\(\displaystyle f(x)\, =\, \log(4)\, \dfrac{x\, +\, 6}{(x\, -\, 1)(x\, +\, 1)}\)

Now when I plotted this on the graph to find the domain and here's what I have: http://www.mathway.com/graph/NTQ4NzM

I don't think I've done this correctly to be honest. I'm also using this video to ensure that the graph I'm doing abides by the rules. https://youtu.be/BV0uLKueNUg

Am I doing this properly?

 

[Deleted member 4993]

Here is the problem I need a bit of help with.

4. Find the domain of the logarithmic function: \(\displaystyle \, f(x)\, =\, \log_4\, \left(\,\dfrac{x\, +\, 6}{x^2\, -\, 1}\,\right)\)

So I believe I am supposed to solve the problem and I got as far as having the following below:

\(\displaystyle f(x)\, =\, \log(4)\, \dfrac{x\, +\, 6}{(x\, -\, 1)(x\, +\, 1)}\)

Now when I plotted this on the graph to find the domain and here's what I have: http://www.mathway.com/graph/NTQ4NzM

I don't think I've done this correctly to be honest. I'm also using this video to ensure that the graph I'm doing abides by the rules. https://youtu.be/BV0uLKueNUg

Am I doing this properly?

What did you conclude? What is/are the domain/s?

 

[ksdhart]

I believe Denis' hint can get you where you want to go, but I'll suggest an alternate method that I personally find easier. But first, a slight correction. In your first picture, the problem as stated is:

\(\displaystyle f\left(x\right)=log_4\left(\frac{x+6}{x^2-1}\right)\)

But then in your next picture, when you re-write the problem and break apart the denominator into two fractions, you write:

\(\displaystyle f\left(x\right)=log\left(4\right)\left(\frac{x+6}{x^2-1}\right)\)

But those two are not the same problem. In the original problem, the 4 is a subscript, which means you would take the log base 4 of the expression of x. In your re-typed problem, the 4 is in parentheses, meaning you would take the log base 10 of 4, then multiply that result by the expression of x.

That distinction being cleared up, for this problem, the fact that they specify a base for the logarithm is actually irrelevant, because the problem only asks for the domain of f(x). No matter what the base is, recall that a logarithm of some function g(x) is only defined if g(x) is positive. Given that, what can you say about the domain of your function f(x)?

 

[stapel]

4. Find the domain of the logarithmic function: \(\displaystyle \, f(x)\, =\, \log_4\, \left(\,\dfrac{x\, +\, 6}{x^2\, -\, 1}\,\right)\)

The above means "the log, base four, of (the fractional expression)".

So I believe I am supposed to solve the problem and I got as far as having the following below:

\(\displaystyle f(x)\, =\, \log(4)\, \dfrac{x\, +\, 6}{(x\, -\, 1)(x\, +\, 1)}\)

The above means "the log [base ten?] of four, multiplied by (the fractional expression)". Are these the same?

Now when I plotted this on the graph to find the domain and here's what I have: http://www.mathway.com/graph/NTQ4NzM

You have graphed something which is defined for negative values of the input. (For instance, when x = -7, the fractional expression evaluates to -1/48.) Are logs defined for negative inputs? So can this graph possibly be the graph of the original function?

 

[Fufufuwie]

The above means "the log, base four, of (the fractional expression)".


The above means "the log [base ten?] of four, multiplied by (the fractional expression)". Are these the same?


You have graphed something which is defined for negative values of the input. (For instance, when x = -7, the fractional expression evaluates to -1/48.) Are logs defined for negative inputs? So can this graph possibly be the graph of the original function?

Okay so that's a definite no and I did the graph wrong. Hmmm okay so when you say, "are these the same?" did you mean the polynomial I just factored? -> (x-1)^2

 

[Fufufuwie]

What did you conclude? What is/are the domain/s?

I think the domains are (-∞,-1)u(1,∞)

I don't think this is right because isn't a log function supposed to be positive?

 

[Fufufuwie]

Typing out loud(!)...
k = f(x), u = (x+6) / (x^2-1)

4^u = k
4 = k^(1/u)

If you like it, say nothing;
if you don't, say "get lost"

I'm sorry but I am still a bit confused. I went ahead and plugged everything in and I received the following:

4^x+6 = f(x)

4 = f(x)^1/(x+6/x^2 -1)

 

[stapel]

4. Find the domain of the logarithmic function: \(\displaystyle \, f(x)\, =\, \log_4\, \left(\,\dfrac{x\, +\, 6}{x^2\, -\, 1}\,\right)\)

The above means "the log, base four, of (the fractional expression)".

So I believe I am supposed to solve the problem and I got as far as having the following below:

\(\displaystyle f(x)\, =\, \log(4)\, \dfrac{x\, +\, 6}{(x\, -\, 1)(x\, +\, 1)}\)

The above means "the log [base ten?] of four, multiplied by (the fractional expression)". Are these the same?

Okay so that's a definite no and I did the graph wrong. Hmmm okay so when you say, "are these the same?" did you mean the polynomial I just factored? -> (x-1)^2

I stated, in words, what your two expressions meant. I then asked if those expressions were the same. No, that did not mean "are you sure that you factored the denominator of the rational expression correctly?" It meant "are you sure that these two descriptions describe the same thing?"

Do they describe the same thing? Does "the log, base four, of a rational expression" mean the same thing as "the product of that rational expression and a constant, where that constant is the log, base ten, of four"? (Hint: No, they can't possibly be the same thing.)

I think the domains are (-∞,-1)u(1,∞)

I don't think this is right because isn't a log function supposed to be positive?

When I stated the following:

You have graphed something which is defined for negative values of the input. (For instance, when x = -7, the fractional expression evaluates to -1/48.) Are logs defined for negative inputs? So can this graph possibly be the graph of the original function?

...this was meant to point out that, since logs cannot take negative arguments, your graph could not possibly be that of a logarithmic function. So, no, this is not correct.

Try working with the original function. Hint: Change-of-base formula. :wink: