Answer
$x=\dfrac{3}{2}=1.5$
Work Step by Step
Recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
Applying this definition to our equation, we get:
$8^{\frac{1}{3}}=2x-1$
$2=2x-1$ (because $2^3=8$)
$3=2x$
$x=\frac{3}{2}=1.5$
We confirm that the answer works:
$\log_{8} (2\cdot 1.5-1)=\frac{1}{3}$
$\log_{8} (3-1)=\frac{1}{3}$
$\log_{8} (2)=\frac{1}{3}$
$\frac{1}{3}=\frac{1}{3}$
