# Solving logarithmic equations

1. Apr 13, 2014

### TheRedDevil18

1. The problem statement, all variables and given/known data

Determine at which points the graphs of the given pair of functions intersect:

f(x) = 3x and g(x) = 2x2

2. Relevant equations

3. The attempt at a solution

I know I have to equate and solve for x so I converted them to logarithms

log3x = log2x2

Don't know if that's right, but I am stuck here, do I use the change of base formula ?

2. Apr 13, 2014

### Dick

With that subscript in there not clear what you mean by "converted them to logarithms". The correct thing to do is to try to solve the equation log(f(x))=log(g(x)). $\log 3^x=\log 2^{x^2}$. The 'log' can be any base you like. Just use the rules of logarithms to solve that equation.

3. Apr 13, 2014

### Staff: Mentor

You cannot just replace exponentials by logarithms, it won't work. There is a way to solve it, but then your steps have to be valid transformations.
That is a good idea, you can do it with the exponentials as well.

4. Apr 13, 2014

### TheRedDevil18

log3x = log2x2

log3x = 2log2x

Using the change of base formula

log2x/log23 = 2log2x............stuck here

5. Apr 13, 2014

### Staff: Mentor

2x2 means 2(x2), not (2x)2, your first step does not work.

What is log(3x) simplified?

6. Apr 14, 2014

### TheRedDevil18

xlog3 = x^2log2

log3 = xlog2

x = log3/log2

Is that correct ?, also why is the base 10 ?, I thought it was 3 and 2 respectively

7. Apr 14, 2014

### Dick

That's part of it. The base doesn't have to be 10. If you take ratio log(3)/log(2) in any base you'll get the same number. Can you say why? More importantly, there is another solution. What is it?

8. Apr 14, 2014

### TheRedDevil18

I think the other solution should be x = 0 as well ?, I'm not too sure about why you get the same number, a bit confused, can you explain that please ?

9. Apr 14, 2014

### HallsofIvy

Starting from $3^x= 2^{x^2}$, you can take the logarithm to any base, "10", "e", whatever, and get $log(3^x)= x log(3)= log(2^{x^2})= x^2 log(2)$. If x is not 0, you can divide both sides by x log(2).