Solve the following logarithm problem for the positive solution for a. log_(64)x=-(4)/(3) Answer Attemptiout of square

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