Find the exact value of x Logarithms

Homework Statement

Find the exact value of x if:

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

The Attempt at a Solution

3lg3xlg3=4lg4xlg4
(xlg3)/(xlg4)=(4lg4)/(3lg3)
xlg3-lg4=(4lg4)(3-(lg3))
xlg(3/4)=(4lg4)(3lg(1/3))

The Attempt at a Solution

ehild
Homework Helper

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

Last edited:
I like Serena
Homework Helper

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

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

ehild
Homework Helper

Exactly! Well done!

ehild