# Exponential Equation on Paper

So, if we start with logarithm ln(2) assuming a base of 10, how would I solve this without a calculator?

ln(2)
10^x = 2

And I get stuck there without a calculator. I could ..

ln(10^x) = ln(2)
x ln(10) = ln(2)
x = ln(2)

But that gets me back to where I started.
So, how would I go about solving 10^x = 2 without a calculator? Using a sort of guessing method, I can't really think of what 10^.1 and 10^.2 would be without going off paper.

I just chose a very basic problem to start with just to get the idea. This is just for myself because I'm curious of learning how to accomplish it. Where this came up was doing things like log(3/2) to find the probability of a (semi) random number starting with 2. I came up with 10^x = 1.5 which wasn't so helpful.

Of course, I plug these into a calculator and get them instantly, but by goal is to do without if it's not extremely difficult.

Thanks

Integral
Staff Emeritus
Gold Member
BC we used a slide rule or math tables.

BC= Before calculators.

Mark44
Mentor
So, if we start with logarithm ln(2) assuming a base of 10, how would I solve this without a calculator?
The base-10 logarithm of a number x is usually written as log(x), not ln(x). ln is always used to mean the natural logarithm, the log with e as the base.
ln(2)
10^x = 2
What I think you're trying to say is this:
log(2) = x
2 = 10x

These two equations are equivalent, meaning whatever value of x works in one equation, also works in the other

I don't know why you would want to attempt to find x without using a calculator. It can be done, but it would take a lot to explain how to do this.

As an alternative, and assuming you are allowed to use a calculator that has a 10x button, you can use an approximation technique to solve for x.

If x = 0, 10x = 1 -- too small, so try a larger value of x.
If x = 1, 10x = 10 -- too big, so try a smaller value
If x = .5, 10x = √10 ≈ 3.162 - too big
If x = .25, 10x = ##\sqrt[4]{10}## ≈ 1.778

and so on.
And I get stuck there without a calculator. I could ..

ln(10^x) = ln(2)
x ln(10) = ln(2)
x = ln(2)

But that gets me back to where I started.
So, how would I go about solving 10^x = 2 without a calculator? Using a sort of guessing method, I can't really think of what 10^.1 and 10^.2 would be without going off paper.

I just chose a very basic problem to start with just to get the idea. This is just for myself because I'm curious of learning how to accomplish it. Where this came up was doing things like log(3/2) to find the probability of a (semi) random number starting with 2. I came up with 10^x = 1.5 which wasn't so helpful.

Of course, I plug these into a calculator and get them instantly, but by goal is to do without if it's not extremely difficult.

Thanks

Well, this isn't for anything specific. It was more for my entertainment. I guess I'll just stick to using a calculator. :P