Finding the Constant k in an Exponential Function with Limited Information

[dragonblood]

I have a problem with an exponential function. I am wondering if an exact solution is possible, or if I have to write the solution as a logarithm of an unknown.

A formula says that $$E(z)=E(0)^{-kz}$$, where E is light intensity and z is depth in water. My objective is to find the constant k. I also know that $$E(3)=0.01E(0)$$.

I have tried to solve for k in the following way:

$$E(3)=0.01E(0)$$

$$E(3)=100E(3)^{-3k}$$
$$\ln |0.01E(3)|=-3k \ln|E(3)|$$

I realize that all values except for k is a constant, however, I do not know the value of E, and my question is: Are there any ways to eliminate E(3) from the equation, leaving k=numerical constant?

-dragonblood

 

[sylas]

I have a problem with an exponential function. I am wondering if an exact solution is possible, or if I have to write the solution as a logarithm of an unknown.

A formula says that $$E(z)=E(0)^{-kz}$$, where E is light intensity and z is depth in water. My objective is to find the constant k. I also know that $$E(3)=0.01E(0)$$.

I have tried to solve for k in the following way:

$$E(3)=0.01E(0)$$

$$E(3)=100E(3)^{-3k}$$
$$\ln |0.01E(3)|=-3k \ln|E(3)|$$

I realize that all values except for k is a constant, however, I do not know the value of E, and my question is: Are there any ways to eliminate E(3) from the equation, leaving k=numerical constant?

-dragonblood

The formula you have is Beer's law, and E(z) is the intensity of light at a given depth. E₀ is the input intensity, or intensity at depth 0.

In your post, you have omitted the all important "e". The relation would normally be given as follows:

$$E(z) = E_0 e^{-kz}$$​

You should be able to solve this for k, given E(3) = 0.01 E(0).

Cheers -- sylas

 

[dragonblood]

Thanks!