Natural Log and Exponential

In summary, the conversation discusses how to solve for K in the equation ln(1-4k)=-6k by taking the e of both sides. It is mentioned that k=0 is a solution and that an advanced topic, the LambertW-Function, can be used to find another solution. It is also noted that the answer is 0.1457 m/s and that the solutions can be visualized by plotting the functions.
  • #1

Homework Statement


Trying to solve for K:

ln(1-4k)=-6k


Homework Equations





The Attempt at a Solution


I know that need to take the e of both side, to get 1-4k=e^-6k, but I cannot remember any properties of e to allow me to remove the k from the exponent. Any tips is appreciated.
 
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  • #2
CursedAntagonis said:

Homework Statement


Trying to solve for K:

ln(1-4k)=-6k


Homework Equations





The Attempt at a Solution


I know that need to take the e of both side, to get 1-4k=e^-6k, but I cannot remember any properties of e to allow me to remove the k from the exponent. Any tips is appreciated.

Clearly, [tex]k=0[/tex] is a solution.
to find the another solution you need to use an advanced topic:
http://mathworld.wolfram.com/LambertW-Function.html
 
  • #4
You have to solve it numerically. You can't solve for k algebraically.

You can visualize the solutions by plotting the functions y=1-4k and y=e^-6k and seeing where they intersect.
 
  • #5
CursedAntagonis said:
the answer is actually 0.1457 m/s.

m/s?? :smile:
 

1. What is the difference between natural log and exponential functions?

Natural log and exponential functions are inverse operations of each other. The natural log function, denoted as ln(x), is the inverse of the exponential function, denoted as e^x. This means that for any value of x, ln(e^x) = x and e^(ln(x)) = x.

2. How are natural log and exponential functions used in real life?

Natural log and exponential functions are commonly used in finance, biology, and physics. In finance, compound interest is modeled using exponential functions. In biology, population growth and decay can be described using exponential functions. In physics, radioactive decay and certain chemical reactions can be modeled using natural log and exponential functions.

3. What is the domain and range of natural log and exponential functions?

The domain of natural log and exponential functions is all real numbers. However, the range of natural log functions is limited to all real numbers greater than 0, while the range of exponential functions is all real numbers.

4. Can natural log and exponential functions be graphed?

Yes, both natural log and exponential functions can be graphed. The graph of a natural log function is a curve that starts at the point (1,0) and increases as x increases. The graph of an exponential function is a curve that starts at the point (0,1) and increases rapidly as x increases.

5. How are natural log and exponential functions related to the number e?

The number e is the base of both natural log and exponential functions. It is an irrational number with a value of approximately 2.71828. The natural log function is the inverse of the exponential function with base e, meaning that e^x and ln(x) cancel each other out when used together. Additionally, the derivative of e^x is e^x, and the derivative of ln(x) is 1/x, making e the natural base for calculus and many other mathematical applications.

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