Exponential Growth and Logarithmic Growth

[austinv]

I'm a very conceptual person, and I've been reading about exponential and logarithmic growth but don't fully have the kind of conceptual grasp on the two and how they differ that I'd like, so I"m curious:

What is the difference between exponential growth and logarithmic growth and what causes this difference?

Also, according to Wikipedia: "In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative." What does it actually mean for ex to be its own derivative? Does it mean that ex always equals e no matter what x is... I don't think so?

And one more related question: in photography, 18% reflectance (18% gray) is said to be perceptually equivalent to 50% of the brightness of 100% reflectance (full white) because our visual system's response to brightness is logarithmic. But my question is what is the math behind how 18% is an equal number of steps between black and white?

Thank you so much!

 

[mathman]

Exponential function is e^x. It has the property that it is equal to its derivative, i.e. d(e^x)/dx = e^x.

Exponential growth means the growth function looks like y ~ e^x. Logarithmic growth is y ~ ln(x). The are related by the fact that y = e^x is the same as x = ln(y).

Photog question - I have no idea.

 

[HallsofIvy]

Also, according to Wikipedia: "In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative." What does it actually mean for ex to be its own derivative? Does it mean that ex always equals e no matter what x is... I don't think so?

Do you know what a derivative is? If not, it is, roughly the rate of change of a function. It can be shown that the rate of change of the function f(x)= a^x is proportional to a^x. "e" has the property that constant of proportionality is 1.

And one more related question: in photography, 18% reflectance (18% gray) is said to be perceptually equivalent to 50% of the brightness of 100% reflectance (full white) because our visual system's response to brightness is logarithmic. But my question is what is the math behind how 18% is an equal number of steps between black and white?

It's not math, it's physiology. Experimental evidense shows that f 18% of a light we had been looking at enters our eyes, we perceive it as "half as bright". While I'm no physiologist, I have read that the more light (or other impulse) strikes our optic nerves (or other nerves), the less they react to each increase in strength.

Thank you so much![/QUOTE]