Answer
$x=\dfrac{4\sqrt{3}}{3}\approx 2.3094$
Work Step by Step
Recall the power property of logarithms (pg. 462):
$\log_b{m^n}=n\log_b{m}$
Applying this property, we get:
$\log_3{\left(2^4\right)}-\log_3{x^2}=1$
Next, recall the quotient property of logarithms (pg. 462):
$\log_b{\frac{m}{n}}=\log_b{m}-\log_b{n}$
Applying this property to our last equation, we get:
$\log_3{\left(\frac{2^4}{x^2}\right)}=1$
Now, recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
Applying this definition, we get:
$3^1=\dfrac{2^4}{x^2}$
$3(x^2)=2^4$
$3x^2=16$
$x^2=\dfrac{16}{3}$
$x=\sqrt{\frac{16}{3}}$
$x=\dfrac{4}{\sqrt{3}}$
$x=\dfrac{4\sqrt{3}}{3}\approx 2.3094$
We confirm that the answer works:
$4\log_3 2-2\log_3 2.3094=1$
$4(0.63)-2(0.76)=1$
$1=1$
