Exponential Equation on Paper

  • Thread starter Xaotique
  • Start date
  • #1
3
0
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
 

Answers and Replies

  • #2
Integral
Staff Emeritus
Science Advisor
Gold Member
7,201
56
BC we used a slide rule or math tables.

BC= Before calculators.
 
  • #3
34,687
6,394
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
 
  • #4
3
0
Well, this isn't for anything specific. It was more for my entertainment. I guess I'll just stick to using a calculator. :P
 

Related Threads on Exponential Equation on Paper

Replies
17
Views
695
Replies
3
Views
7K
Replies
3
Views
6K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
888
  • Last Post
Replies
5
Views
7K
Top