Natural Logarithm Manupulations

In summary, the equation xln(2x+1)-x+\frac{1}{2}ln(2x+1) = \frac{1}{2}(2x+1)ln(2x+1)-x can be simplified by factoring out the common term ln(2x+1) from both sides, resulting in xln(2x+1)+\frac{1}{2}ln(2x+1) = \frac{1}{2}(2x+1)ln(2x+1). This simplification helps to understand the relationship between the two sides of the equation and makes it easier to solve.
  • #1
bob1182006
492
1

Homework Statement


[tex]xln(2x+1)-x+\frac{1}{2}ln(2x+1) = \frac{1}{2}(2x+1)ln(2x+1)-x[/tex]

Homework Equations


[tex]ln(x^a) = aln(x), ln(xy) = ln(x) + ln(y), ln(\frac{x}{y}) = ln(x) - ln(y)[/tex]

The Attempt at a Solution



I have no idea how you can go from [itex] xln(2x+1)-x+\frac{1}{2}ln(2x+1)[/itex] to [itex]\frac{1}{2}(2x+1)ln(2x+1)-x[/itex] could someone point me in the right direction?

I know both sides have the -x term, so the only change takes place in [itex]xln(2x+1)+\frac{1}{2}ln(2x+1) = \frac{1}{2}(2x+1)ln(2x+1)[/itex]
 
Last edited:
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  • #2
factor out the ln(2x+1)
 
  • #3
tim_lou said:
factor out the ln(2x+1)

wow I can't believe I didn't see that, thanks so much I get it now
 

1. What is a natural logarithm?

A natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted by ln(x) and is used to find the exponent to which the base e must be raised to get a certain number x. The value of e is approximately 2.71828.

2. How do you simplify expressions with natural logarithms?

To simplify expressions with natural logarithms, you can use the following properties:
- ln(a*b) = ln(a) + ln(b)
- ln(a/b) = ln(a) - ln(b)
- ln(a^b) = b*ln(a)
- ln(e) = 1
- ln(1) = 0
These properties can help you combine logarithmic expressions and simplify them into a single logarithm.

3. Can natural logarithms be negative?

Yes, natural logarithms can be negative. The only restriction is that the input of the ln function must be a positive number. This means that the output of the function can be any real number, including negative numbers.

4. How do you solve equations with natural logarithms?

To solve equations with natural logarithms, you can use the following steps:
1. Isolate the natural logarithm on one side of the equation.
2. Use the inverse property of logarithms to rewrite the equation in exponential form.
3. Solve for the variable.
4. Check your answer by plugging it back into the original equation.

5. Can natural logarithms be used in real-world applications?

Yes, natural logarithms have many real-world applications. Some examples include calculating the half-life of a radioactive substance, predicting population growth, and determining the rate of decay in electronics. They are also used in finance and economics to calculate compound interest and inflation rates.

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