Question

I need help with solving the log​

Answer 1

(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:

•
`\log\left(\frac ab\right)=\log(a)-\log(b)`

•
`\log(ab)=\log(a)+\log(b)`

•
`\log(a^b)=b\log(a)`

By the first property,

`\log_5\left((r^(\frac98))/(t^(\frac32)x^(\frac58))\right) = \log_5\left(r^(\frac98)\right) - \log_5\left(t^(\frac32)x^(\frac58)\right)`

By the second property,

`\log_5\left(t^(\frac32)x^(\frac58)\right) = \log_5\left(t^(\frac32)\right) + \log_5\left(x^(\frac58)\right)`

By the third property,

`\log_5\left(r^(\frac98)\right)=\frac98\log_5(r)`

`\log_5\left(t^(\frac32)\right)=\frac32\log_5(t)`

`\log_5\left(x^(\frac58)\right)=\frac58\log_5(x)`

Putting everything together, we get the expanded expression

`\log_5\left(\sqrt[8]{(r^9)/(t^(12)x^5)}\right) = \frac98\log_5(r) - \left(\frac32\log_5(t) + \frac58\log_5(x)\right)`

Now just plug in the given values to get

`\log_5\left(\sqrt[8]{(r^9)/(t^(12)x^5)}\right) = \boxed{9.97}`