To solve for [tex]\( x \)[/tex] in the equation [tex]\[ \log_{64} (x) = \frac{1}{2} \][/tex], let's follow these steps:
1. Understand the Definition of Logarithms: The equation [tex]\(\log_{64} (x) = \frac{1}{2}\)[/tex] means that [tex]\(64\)[/tex] raised to the power of [tex]\(\frac{1}{2}\)[/tex] equals [tex]\(x\)[/tex]. In other words,
[tex]\[ 64^{\frac{1}{2}} = x. \][/tex]
2. Interpret the Exponent: The exponent [tex]\(\frac{1}{2}\)[/tex] represents a square root. Therefore, [tex]\( 64^{\frac{1}{2}} \)[/tex] is the same as the square root of 64.
3. Calculate the Square Root: We need to find the number which, when squared, gives 64. That number is:
[tex]\[ \sqrt{64} = 8. \][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = 8. \][/tex]
So, the solution to [tex]\(\log_{64} (x) = \frac{1}{2}\)[/tex] is [tex]\( x = 8. \)[/tex]
