Hi
i need to find x in 9^(2x-6)=6^(8x-5)
So what i did was:
3^(4x-12)=6^(8x-5)
log3(3)^(4x-12)=log3(6)^(8x-5)
4x-12=(8x-5)log3(6)
now i'm stuck. and i actually feel that the whole thing is wrong. help, pls
9^(2x-6)=6^(8x-5)
(2x-6) log9 = (8x-5)log6
(2log9)x - 6log9 = (8log6)x - 5log6
x = (6log9 - 5log6)/(2log9 - 8log6)
you can massage that a few ways, but that's the idea
thanks!
Hi! Let's go through the steps together to solve the equation 9^(2x-6) = 6^(8x-5).
You're on the right track by taking the logarithm of both sides of the equation. However, there seems to be a mistake in your approach. I'll walk you through the correct steps:
Step 1: Start with the equation 9^(2x-6) = 6^(8x-5).
Step 2: Take the logarithm of both sides of the equation. You can use any base for the logarithm, but let's use the natural logarithm (ln).
ln(9^(2x-6)) = ln(6^(8x-5))
Step 3: Apply the exponent rule of logarithms, which states that ln(a^b) = b * ln(a). This allows us to bring the exponent down as a coefficient.
(2x-6)*ln(9) = (8x-5)*ln(6)
Step 4: Distribute the logarithms:
2x*ln(9) - 6*ln(9) = 8x*ln(6) - 5*ln(6)
Step 5: Simplify by gathering the terms with x on one side and the constant terms on the other side:
2x*ln(9) - 8x*ln(6) = 6*ln(9) - 5*ln(6)
Step 6: Factor out x from the left side:
x * (2ln(9) - 8ln(6)) = 6ln(9) - 5ln(6)
Step 7: Finally, divide both sides by (2ln(9) - 8ln(6)) to solve for x:
x = (6ln(9) - 5ln(6)) / (2ln(9) - 8ln(6))
That's it! You have found the solution for x in the equation.