Need help with graphing natural base exponential and logarithmic functions BY HAND

[GrannySmith]

I don't understand how one is supposed to graph NATURAL BASE exponential functions and logarithmic functions by hand? How the heck is this even remotely possible.

For example, f(x) = (1/4)e^x

Alright, I get that there is a point at (0,.25) but aside from that, I don't have the slightest CLUE on how the rest of the graph looks, only the general shape. I'd need a calculator to even approximate them. If any of you guys have some tips or tricks to help me out, I'd love to know.

 

[Ishuda]

I don't understand how one is supposed to graph NATURAL BASE exponential functions and logarithmic functions by hand? How the heck is this even remotely possible.

For example, f(x) = (1/4)e^x

Alright, I get that there is a point at (0,.25) but aside from that, I don't have the slightest CLUE on how the rest of the graph looks, only the general shape. I'd need a calculator to even approximate them. If any of you guys have some tips or tricks to help me out, I'd love to know.

Were I doing the problem, I would assume that a calculator could be used to get the values, i.e. (1, 0.68) for example. You just couldn't use a calculator to do the graph.

Of course if you have a teacher that expects you to find a table of natural logarithms and use those rather than a calculator, well good luck.

 

[Ishuda]

I don't understand how one is supposed to graph NATURAL BASE exponential functions and logarithmic functions by hand? How the heck is this even remotely possible.

For example, f(x) = (1/4)e^x

Alright, I get that there is a point at (0,.25) but aside from that, I don't have the slightest CLUE on how the rest of the graph looks, only the general shape. I'd need a calculator to even approximate them. If any of you guys have some tips or tricks to help me out, I'd love to know.

BTW: I forgot to ask over what range. If it were from say, 0 to 0.1 you could use the approximation
e^x ~ 1 + x + x²/2
and, of course you could use e^x = (e^x/2)² to extend the interval and continue from there.

 

[Steven G]

I don't understand how one is supposed to graph NATURAL BASE exponential functions and logarithmic functions by hand? How the heck is this even remotely possible.

For example, f(x) = (1/4)e^x

Alright, I get that there is a point at (0,.25) but aside from that, I don't have the slightest CLUE on how the rest of the graph looks, only the general shape. I'd need a calculator to even approximate them. If any of you guys have some tips or tricks to help me out, I'd love to know.

As you stated you know the general shape of the graph so all you need is two points. I would always use the x values that make the power equal to 0 and 1 (as Ishuda suggested). In your case that is when x=0 and x=1. The y values will be (1/4) and (1/4)e. So label your y-axis by one-quarters-- 1(1/4) or 1/4, 2(1/4) or 1/2, 3(1/4) or 3/4. 4(1/4) or 1,,,. Now since e~2.718, we know that e(1/4) ~ 2.718(1/4) which is in between 2(1/4) and 3(1/4)-actually closer to 3(1/4). Now plot those two points and graph your exponential function.