(See screenshot for the first step that I *would* include, but apparently the website thinks it contains profanity...)
Now recall the following properties of logarithms:
• [tex]\log\left(\frac ab\right)=\log(a)-\log(b)[/tex]
• [tex]\log(ab)=\log(a)+\log(b)[/tex]
• [tex]\log(a^b)=b\log(a)[/tex]
By the first property,
[tex]\log_5\left(\dfrac{r^{\frac98}}{t^{\frac32}x^{\frac58}}\right) = \log_5\left(r^{\frac98}\right) - \log_5\left(t^{\frac32}x^{\frac58}\right)[/tex]
By the second property,
[tex]\log_5\left(t^{\frac32}x^{\frac58}\right) = \log_5\left(t^{\frac32}\right) + \log_5\left(x^{\frac58}\right)[/tex]
By the third property,
[tex]\log_5\left(r^{\frac98}\right)=\dfrac98\log_5(r)[/tex]
[tex]\log_5\left(t^{\frac32}\right)=\dfrac32\log_5(t)[/tex]
[tex]\log_5\left(x^{\frac58}\right)=\dfrac58\log_5(x)[/tex]
Putting everything together, we get the expanded expression
[tex]\log_5\left(\sqrt[8]{\dfrac{r^9}{t^{12}x^5}}\right) = \dfrac98\log_5(r) - \left(\dfrac32\log_5(t) + \dfrac58\log_5(x)\right)[/tex]
Now just plug in the given values to get
[tex]\log_5\left(\sqrt[8]{\dfrac{r^9}{t^{12}x^5}}\right) = \boxed{9.97}[/tex]
