How Do I Rewrite y = 5x as 3kx?

[the_awesome]

Homework Statement

Rewrite y = 5^x in the form 3^kx

Homework Equations

no idea

The Attempt at a Solution

I can't really attempt this because i don't have a clue where to start. I've looked in all my maths books and it says nothing on how to do this. Our unit is partially based on inverses and differentation so I am guessing its something to do with that.

By guessing and typing random numbers into the calculator, i know that k is equal to 1.465

example :

5² = 25
3^{1.465 x 2} = 25

thats all i can gather, but how do i show my working out/correct method? :(

 

[rock.freak667]

well if you want 3^(kx)=5^x

take logs to find k.

 

[the_awesome]

how do i do that?

 

[Vagrant]

Have you studied logarithms?
If you have, use log a^b = b log a to find the value of k.

 

[lanedance]

do you know about logarithmic functions?

take the log of both sides and from the log laws, you can show log(y^k) = k.log(y)

 

[the_awesome]

Have you studied logarithms?
If you have, use log a^b = b log a to find the value of k.

so u mean like...

(let a = 5)
(let b = 2)

log 5² = 2 log 5
= 1.39

but, the value of k should be 1.465.


what about...
5^x = 3^kx
0 = 5^x - 3^kx
0 = 5 -3^k
3^k = 5
k = (log 5) divided by (log 3)

therefore, k = 1.464973521

so...5x = 3^1.465x

 

[HallsofIvy]

so u mean like...

(let a = 5)
(let b = 2)

log 5² = 2 log 5
= 1.39

but, the value of k should be 1.465.


what about...
5^x = 3^kx
0 = 5^x - 3^kx
0 = 5 -3^k

No, that does not follow. You cannot just "cancel" exponents like that.
You can, directly from your original equation say that
5^x= 3^kx= (3^k)^x and NOW take the "x^th" root of both sides: 5= 3^k.

3^k = 5
k = (log 5) divided by (log 3)

therefore, k = 1.464973521

so...5x = 3^1.465x

That works, with my correction to your reasoning. What people were suggesting you do is simpler: just take the logarithm of both sides of the original equation:
log(5^x)= log(3^kx)
x log(5)= kx log(3)
Since this is to be true for all x, you can take x non-zero and divide both sides by x:
log(5)= k log(3) so k= log(5)/log(3).

 

[the_awesome]

ah i see, thanks for the help!