solve
4^-x=64
Please help
-x log 4 = log 64
-x = log 64/log 4 = 3
x = -3
To solve the equation 4^(-x) = 64, we can use logarithms. Here's the step-by-step process:
1. Take the logarithm of both sides of the equation. Since the base of the exponent is 4, we will use the logarithm base 4.
log₄(4^(-x)) = log₄(64)
2. Apply the logarithmic property which states that logₐ(a^b) = b * logₐ(a). This allows us to bring the exponent down as a coefficient:
-x * log₄(4) = log₄(64)
3. Simplify the logarithm of 4. The logarithm of the base raised to itself is equal to 1:
-x * 1 = log₄(64)
4. Simplify further:
-x = log₄(64)
5. Determine the value of log₄(64). This can be evaluated as log₄(64) = log(64) / log(4). Using a calculator, we can find:
log₄(64) ≈ 3
6. Substitute the value back into the equation:
-x = 3
7. Solve for x by multiplying both sides of the equation by -1:
x = -3
Therefore, the solution to the equation 4^(-x) = 64 is x = -3.