Is the rewritten form of ln(x2) valid?

In summary, when rewriting a ln function, it is important to ensure that the domain remains the same. The example provided shows that the rewritten form ln(x^2) = 2 ln(|x|) is valid, as ln(x^2) and 2 ln(x) have the same domain. The difference in the graphs is due to the limitations on the values of the arguments for logarithms.
  • #1
mindheavy
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I'm reading back over a calculus book getting ready for an exam and I'm seeing a note that I don't understand.

It says to make sure, when rewriting a ln function that the domain is the same, then it provides an example of when it's not the same, yet says nothing more. Is this rewritten form valid?

https://dl.dropbox.com/u/15809883/ln.png
 
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  • #2
ln(x^2) = 2 ln(|x|)

As sqrt(x^2)=|x|
 
  • #3
Ah yes, I had forgotten about that, thank you.
 
  • #4
The difference in the graphs is entirely due to the domains of the two different functions.

ln(x2) is defined for all real x ≠ 0.
2 ln(x) is defined only for x > 0.

The rules for logarithms contain limitations on the values of the arguments. For example, ln(a*b) = ln(a) + ln(b), where a > 0 and b > 0. Note that it is possible for ln(a * b) to be defined even though the right side is undefined. This can happen when both a and b are negative.
 

FAQ: Is the rewritten form of ln(x2) valid?

What is the natural log function?

The natural log function, denoted as ln(x), is the inverse of the exponential function with a base of e. It is a mathematical function that represents the logarithm of a number with respect to the base e, which is approximately equal to 2.71828.

What is the significance of the base e in the natural log function?

The base e, also known as Euler's number, is a fundamental constant in mathematics that appears in many natural phenomena, such as population growth and compound interest. It is also the only number for which the derivative of the natural log function is equal to 1.

What is the domain and range of the natural log function?

The domain of the natural log function is all positive real numbers, as the input value must be greater than 0. The range of the function is all real numbers, as the output can be any real number.

How is the natural log function used in science?

The natural log function is commonly used in various scientific fields, such as biology, chemistry, and physics. It is used to model natural processes that exhibit exponential growth or decay, such as radioactive decay and population growth. It is also used in statistical analysis to transform data that is skewed towards higher values.

What are some important properties of the natural log function?

Some important properties of the natural log function include the fact that ln(1) = 0, ln(e) = 1, and ln(xy) = ln(x) + ln(y). It also has a unique derivative of 1/x, and its graph is a smooth, continuously increasing curve. Additionally, the natural log function is the inverse of the exponential function, which means that ln(e^x) = x and e^(ln(x)) = x.

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