Intersecting Functions: Solving Logarithmic Equations

[TheRedDevil18]

Homework Statement

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

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

Homework Equations

The Attempt at a Solution

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

log₃^x = log₂^x2

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

 

[Dick]

Homework Statement

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

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

Homework Equations

The Attempt at a Solution

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

log₃^x = log₂^x2

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

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.

 

[mfb]

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.

do I use the change of base formula ?

That is a good idea, you can do it with the exponentials as well.

 

[TheRedDevil18]

log3^x = log2^x2

log3^x = 2log2^x

Using the change of base formula

log2^x/log2³ = 2log2^x...stuck here

 

[mfb]

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

What is log(3^x) simplified?

 

[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

 

[Dick]

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

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?

 

[TheRedDevil18]

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?

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 ?

 

[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).