Exponential Functions

Question

The intensity of light decreases by 4% for each meter that it descends below the surface of the water. At what depth is the intensity of light only 20% of that at the surface?

Answer 1

m - number of meters under surface of water when intensity of light is 20%

1(1 - 0.04)^m = 0.2

m = lg(0.2)/lg(0.96) ≅ 39.43 (m)

Answer 2

If we know that each meter the intensity of light decreases by 4 percent then that means the light 1 meter down is 96 percent as strong as the light above. So in this case, if we were x meters down, the intensity of the light would be 0.96^x. In order to find the point at which the intensity is only 20 percent, we need to find the x value such that 0.96^x = 0.20.

To solve this equation, we need to take log_0.96 of both sides.

So, in this case, we get that x = log_0.96(0.20). If your calculator allows you to input logs of any base, you're done. If it doesn't, you can do the alternative step listed below:

Alternatively, if you remember your log rules, x = log_0.96(0.20) can be written as x = log_a(0.20)/ log_a(0.96) where a is any positive real number. So if we want to use ln (natural log, log base e), x could just as easily be written as ln(0.20)/ln(0.96).

In either case, the answer is approximately 39.43 meters.