To solve the equation [tex]\(\log_{64} 2 = x\)[/tex], follow these steps:
1. Understand the Logarithmic Equation: The given equation [tex]\(\log_{64} 2 = x\)[/tex] can be rewritten in its exponential form. By definition, [tex]\(\log_b a = c\)[/tex] means [tex]\(b^c = a\)[/tex]. Thus, [tex]\(\log_{64} 2 = x\)[/tex] means:
[tex]\[
64^x = 2
\][/tex]
2. Express 64 as a Power of 2: Recognize that 64 can be expressed as a power of 2. Specifically:
[tex]\[
64 = 2^6
\][/tex]
So we can rewrite the equation [tex]\(64^x = 2\)[/tex] as:
[tex]\[
(2^6)^x = 2
\][/tex]
3. Simplify the Exponential Equation: Use the properties of exponents to simplify the left side of the equation:
[tex]\[
2^{6x} = 2^1
\][/tex]
4. Set the Exponents Equal to Each Other: Since the bases are the same (both are 2), the exponents must be equal:
[tex]\[
6x = 1
\][/tex]
5. Solve for [tex]\(x\)[/tex]: Divide both sides of the equation by 6 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{6}
\][/tex]
Thus, the solution to the equation [tex]\(\log_{64} 2 = x\)[/tex] is:
[tex]\[
x = \frac{1}{6}
\][/tex]
So, the correct option is:
[tex]\[
x = \frac{1}{6}
\][/tex]
