Thread: How do you find the intersections between a line and a logarithm?

For example, if you were given the line y = 3x and the logarithm y = log(x) + 5, to find the point(s) of intersection between them, you'd have to solve the equation 3x = log(x) + 5. I have no idea how to solve that for x.

Online graphing calculators show that there are actually two intersections in this example: (0.0001, 0.9897), (1.7475, 5.2424).

Is there an easy solution I'm missing, or do I need to learn something new (heaven forbid )?

Equations like that will typically require the Lambert W function, also known as the product log function, and this is no exception. Your best bet right now is to find the solutions numerically.

Thanks a lot--that's a great website (although it's worth mentioning that for some reason it handles log(x) as ln(x)--if you look at the bottom of the "Input interpretation" box it says "log(x) is the natural logarithm." Still got the answer right for 3x = ln(x) + 5, though).

EDIT: Nevermind ^^^ what's in parenthesis above, I didn't see that I just had to click something to change that.

Unfortunately I know very little about the Lambert W function. Any suggestions for a good explanation of it? (Wikipedia was a bit hard to follow, and I haven't even been to a pre-calc class yet, so I'm not surprised if there isn't). Thanks anyways :)

Originally Posted by MRich520

Thanks a lot--that's a great website (although it's worth mentioning that for some reason

That reason is that Stephen Wolfram, the creator of Mathematica, which is the engine behind WolframAlpha, is a physicist. And physicists often write the natural logarithm as log.

it handles log(x) as ln(x)--if you look at the bottom of the "Input interpretation" box it says "log(x) is the natural logarithm." Still got the answer right for 3x = ln(x) + 5, though).

Unfortunately I know very little about the Lambert W function. Any suggestions for a good explanation of it? (Wikipedia was a bit hard to follow, and I haven't even been to a pre-calc class yet, so I'm not surprised if there isn't). Thanks anyways

The wiki is precisely where I would have pointed you first. You could also check out MathWorld's article.

Okay...looks like I'm in over my head. I have a lot of trouble learning new material off websites as opposed to learning from teacher, so I'll wait to get a better understanding of this area in class. I was just curious about this and if I was missing some kind of algebraic way of solving it. Regardless, thanks for the help.

Originally Posted by MRich520

Okay...looks like I'm in over my head. I have a lot of trouble learning new material off websites as opposed to learning from teacher, so I'll wait to get a better understanding of this area in class. I was just curious about this and if I was missing some kind of algebraic way of solving it. Regardless, thanks for the help.

First define a new function f(x)=3x - log(x) - 5

And now...

try to understand: Newton's method - Wikipedia, the free encyclopedia