Solving for x using logarithms

[Opus_723]

Homework Statement

I tutor math for a couple of high school kids, and usually don't have any problems. Occasionally we run into a problem that takes me a minute, since I haven't actually used a lot of this stuff since I was in high school, but I always figure it out very quickly when that happens.

However, yesterday I came across a problem in the middle of a set of really straightforward logarithm problems, and I couldn't work out how to do it. It was a little embarassing, but I'm more curious than anything. How would one go about solving an equation like this?

30(1.4)^{x} = 30 + 0.4x

Of course, the numbers aren't important, just the general layout, with an x in an exponent and in a term.

No matter how I work it, I can't seem to isolate the x. It seems like there must be some simple approach, especially since all of the other problems were so easy, but I can't think of it.

 

[dalcde]

Newton–Raphson method seems to be the only way.

By the way, x=0 is also a solution to your equation.

 

[clamtrox]

You have to be kind of clever to solve this. Here are possible steps:

1) show there are two solutions
2) show that x=0 is a solution
3) second solution is roughly where 30+0.4x is "small" -> x ~-75 -> 1.4^x ~ 0 so second solution is x≈-75.

In general, you can't solve this kind of equations analytically.

 

[Opus_723]

How can you show that there are two solutions, and that one is where x is "small"?

 

[vela]

Graph the two sides of the equation. You can quickly see there will be two solutions and that one of them is near the x-axis.

 

[Ratch]

Opus 723,

There are many equations and are not amenable to a closed form solution. For instance, cos(x) = x. You can use approximation methods like the Newton-Raphson mentioded earlier, or you can express a term in the expression with a few terms of a fast converging series, and solve the equation that way. Cos(x) in the example I gave can be expressed as a series.

Ratch