solve for x: 8^x=(1/64)^(x-4)
8^x = (1/64)^(x-4),
8^x = (8^-2)(x-4),
8^x = 8^(-2x+8),
If 2 equal numbers have equal bases,their exponents are also equal:
x = -2x+8,
x+2x = 8,
3x = 8,
x = 8/3.
To solve for x in the equation 8^x = (1/64)^(x-4), we can start by simplifying the expression on the right side of the equation.
We can rewrite 1/64 as 2^(-6) because 64 is equal to 2^6. So, we have:
8^x = (2^(-6))^(x-4)
To simplify further, we can apply the rule of exponents which states that (a^b)^c = a^(b * c). Using this rule, we can rewrite the expression on the right as:
8^x = 2^((-6) * (x-4))
Simplifying the exponent on the right:
8^x = 2^(-6x + 24)
Now that both sides of the equation have the same base (2), we can set the exponents equal to each other:
x = -6x + 24
Next, let's solve for x. Move the -6x term to the left side and the 24 term to the right side:
x + 6x = 24
Combining like terms:
7x = 24
Finally, divide both sides of the equation by 7 to solve for x:
x = 24/7
Therefore, x is equal to 24/7.