Solve for x:
8^(6x-3)=(1/16)^(5x-9)
We note that the base of the powers are related to 2, for example, 8=2³, and (1/16)=2-4.
8^(6x-3)=(1/16)^(5x-9)
(2³)^(6x-3) = (2-4)^(5x-9)
Apply the laws of exponentiation,
2^(3(6x-3))=2^(-4(5x-9))
Take log to the base 2 on both sides:
3*(6*x-3)=-4*(5*x-9)
Solve for x to get 45/38.
8^(6x - 3) = (1/16)^(5x - 9),
(2^3)^(6x - 3) = (1/2^4)^(5x - 9),
(2^3)^(6x - 3) = (2^-4)^(5x - 9),
2^(18x - 9) = 2^(-20x + 36),
18x - 9 = -20x + 36,
18x + 20x = 36 + 9,
38x = 45,
x = 45 / 38 = 1.184.
To solve for x in the equation 8^(6x-3) = (1/16)^(5x-9), we can first simplify both sides of the equation by expressing the numbers as powers of 2.
8 can be expressed as 2^3, and 1/16 can be expressed as (1/2)^4.
Rewriting the equation:
(2^3)^(6x-3) = [ (1/2)^4 ]^(5x-9)
Applying the power of a power rule, we can simplify further:
2^(3*(6x-3)) = (1/2)^(4*(5x-9))
Rearranging:
2^(18x-9) = (1/2)^(20x-36)
Now, we can use the fact that if two numbers with the same base are equal, then their exponents must also be equal. Thus, we can set the exponents equal to each other:
18x-9 = 20x-36
Let's solve for x now:
18x - 20x = -36 + 9
-2x = -27
Divide both sides by -2:
x = -27 / -2
Finally, the solution to the equation is:
x = 27/2
So, x is equal to 27/2.