I with a natural log problem?

[empty.soul]

Homework Statement

This is my first time ever posting anything on here...but we just started working with ln? I know that it's the base e? or something like that...
but the problem is this...

1/3=ln(x^2/x-4)

Homework Equations

The Attempt at a Solution

I haven't attempted..i don't know how to do it o.o

 

[ideasrule]

ln is the opposite of "e to the power of", just like how addition is the opposite of subtraction. That is, if you raise e to the power of a certain number, taking the ln of the result gives you back the original number.

I presume you have to solve for x in 1/3=ln(x^2/x-4). I'll get you started: what happens when you raise e to the power of both sides? That is, e^(1/3)=e^(ln(x^2/x-4)), which simplifies to...

 

[kerimek]

I understand that learning new concepts can be overwhelming at first. However, with practice and a deeper understanding of the concept, you will be able to tackle problems involving natural logarithms confidently. To help you get started, here are some steps you can follow to solve this problem:

1. First, let's clarify the problem. The equation given is 1/3 = ln(x^2/x-4). This means that the natural logarithm of a certain value is equal to 1/3. Our goal is to find the value of x that satisfies this equation.

2. To solve for x, we need to isolate it on one side of the equation. We can do this by first multiplying both sides by 3, which will give us 1 = 3ln(x^2/x-4).

3. Next, we can use the property of logarithms that states ln(a/b) = ln(a) - ln(b). This means we can rewrite the equation as 1 = 3ln(x^2) - 3ln(x-4).

4. Using another property of logarithms, ln(a^b) = bln(a), we can simplify further to 1 = ln(x^6) - ln(x-4)^3.

5. Now, we can use the property ln(a) - ln(b) = ln(a/b) again to rewrite the equation as 1 = ln(x^6/(x-4)^3).

6. From here, we can take the inverse of the natural logarithm, which is e^x, on both sides to get e^1 = x^6/(x-4)^3.

7. Simplifying further, we get e = (x^6)/(x-4)^3.

8. We can then cross-multiply to get e(x-4)^3 = x^6.

9. Expanding the left side, we get ex^3 - 12ex^2 + 48ex - 64e = x^6.

10. Rearranging the terms, we get x^6 - ex^3 + 12ex^2 - 48ex + 64e = 0.

11. This is now a polynomial equation that can be solved using various methods, such as factoring or the quadratic formula.

12. Once you have found the values of x that satisfy this equation, you can plug them back into the original