# How Do I Solve This Natural Logarithm Problem?

• empty.soul
In summary, ln is the opposite of "e to the power of" and is used to find the original number when e is raised to a certain power. The task is to solve for x in the equation 1/3=ln(x^2/x-4). To begin, raise e to the power of both sides, which simplifies to e^(1/3)=e^(ln(x^2/x-4)).
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)

## The Attempt at a Solution

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

Last edited:
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...

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

## 1. What is a natural logarithm?

A natural logarithm is a mathematical function that helps us solve problems involving exponential growth and decay. It is often denoted by the symbol "ln" and is the inverse of the exponential function.

## 2. How do I solve a natural log problem?

To solve a natural log problem, you first need to identify the base of the logarithm. Then, you can use the properties of logarithms to rewrite the expression in a simpler form. Finally, you can use your calculator to evaluate the expression and find the solution.

## 3. What are the properties of natural logarithms?

Some of the main properties of natural logarithms include the product rule, quotient rule, and power rule. These rules help us simplify and solve logarithmic expressions.

## 4. What is the difference between a natural logarithm and a common logarithm?

A natural logarithm has a base of "e" (Euler's number), while a common logarithm has a base of 10. This means that natural logarithms are used for exponential functions with a base of "e", while common logarithms are used for exponential functions with a base of 10.

## 5. How are natural logarithms used in real life?

Natural logarithms have many applications in fields such as science, finance, and engineering. They can be used to model population growth, radioactive decay, and compound interest, among other things.

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