Answer
$x\approx 2.1073$
Work Step by Step
We are asked to solve:
$2^{3x-4}=5$
Recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
Applying this definition to our equation (with $b=2, y=3x-4, x=5$), we get:
$\log_{2}{5}=3x-4$
Next, recall the change of base formula (pg. 464):
$\log_{b}{m}=\dfrac{\log_{c}{m}}{\log_{c}{b}}$
Applying this formula to our last equation, we get:
$\log_{2}{5}=3x-4$
$\dfrac{\log_{10}{5}}{\log_{10}{2}}=3x-4$
$\dfrac{\log_{10}{5}}{\log_{10}{2}}+4=3x$
$\dfrac{1}{3} \cdot \dfrac{\log_{10}{5}}{\log_{10}{2}}+\dfrac{4}{3}=x$
$x\approx 2.1073$
We confirm that the answer works:
$2^{3\cdot 2.1073-4}=5$
$2^{2.3219}=5$
$5=5$
