Find the exact value of x Logarithms

[LiHJ]

Homework Statement

Find the exact value of x if:

Homework Equations

(3x)^lg3=(4x)^lg4.

The Attempt at a Solution

3^lg3x^lg3=4^lg4x^lg4
(x^lg3)/(x^lg4)=(4^lg4)/(3^lg3)
x^lg3-lg4=(4^lg4)(3^-(lg3))
x^lg(3/4)=(4^lg4)(3^lg(1/3))

Please help me, I am stuck here!

 

[ehild]

Write both sides of the original equation as power of 10, and compare the exponents (which is the same as taking the lg of both sides)
So you get

lg(3)lg(3x)=lg(4)lg(4x)

that is, (lg3)²+(lg3) lg(x)=(lg4)²+(lg4)lg(x)

Isolate lg(x), use the identity a²-b²=(a-b)(a+b) and simplify. You get lg(x) as the logarithm of a number, from which the exact value of x is found.

ehild

 

[I like Serena]

If x^a=b, then x=b^(1/a).

 

[LiHJ]

Thank you so much echild. The answer is (1/12)

 

[ehild]

Exactly! Well done!

ehild