Question Solve the following logarithm problem for the positive solution for z. log_(256)x=-(3)/(4) Answer Attemptiout of square

To solve the logarithmic equation \( \log_{256} x = -\frac{3}{4} \), we need to convert it to its exponential form. The general form of a logarithmic equation \( \log_b a = c \) can be rewritten in exponential form as \( b^c = a \).<br /><br />Given:<br />\[ \log_{256} x = -\frac{3}{4} \]<br /><br />We rewrite this in exponential form:<br />\[ 256^{-\frac{3}{4}} = x \]<br /><br />Next, we simplify \( 256^{-\frac{3}{4}} \). First, note that \( 256 \) can be expressed as a power of 2:<br />\[ 256 = 2^8 \]<br /><br />So, we have:<br />\[ 256^{-\frac{3}{4}} = (2^8)^{-\frac{3}{4}} \]<br /><br />Using the property of exponents \((a^m)^n = a^{mn}\), we get:<br />\[ (2^8)^{-\frac{3}{4}} = 2^{8 \cdot -\frac{3}{4}} = 2^{-6} \]<br /><br />Now, \( 2^{-6} \) is the same as \( \frac{1}{2^6} \):<br />\[ 2^{-6} = \frac{1}{2^6} = \frac{1}{64} \]<br /><br />Therefore, the positive solution for \( x \) is:<br />\[ x = \frac{1}{64} \]<br /><br />So, the answer is:<br />\[ \boxed{\frac{1}{64}} \]