Solve each equation. Round answers to the nearest hundredth as needed. (1/4)^x=64

Verified step by step guidance

1. Recognize that the equation is an exponential equation of the form \(\left(\frac{1}{4}\right)^x = 64\).
2. Rewrite both sides of the equation with the same base if possible. Note that \(\frac{1}{4}\) can be written as \$4^{-1}\(, and \)64\( can be expressed as a power of 4 since \)64 = 4^3$.
3. Substitute these expressions back into the equation to get \(\left(4^{-1}\right)^x = 4^3\).
4. Use the power of a power property: \(\left(a^m\right)^n = a^{mn}\), so rewrite the left side as \$4^{-x} = 4^3$.
5. Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(-x = 3\). Then solve for \(x\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as (1/4)^x = 64. Solving these requires understanding how to manipulate and isolate the variable in the exponent to find its value.

Properties of Exponents

Properties of exponents, like a^(m) * a^(n) = a^(m+n) and (a^m)^n = a^(mn), help simplify and rewrite expressions. Recognizing equivalent bases allows rewriting both sides of the equation to solve for the exponent.

Logarithms

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms to both sides helps isolate the exponent and solve for the variable.