# write the expression as a single logarithm `1/5ln(x^2+4)-1/2ln(x^2-8)-ln(x-1)`

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We will use the following rule

`a log_b c=log_b c^a`

`1/5ln(x^2+4)-1/2ln(x^2-8)-ln(x-1)=`

`ln root(5)(x^2+4)-ln sqrt(x^2-8)-ln(x-1)`

Now we use this rule

`log_b x-log_b y=log_b x/y`

`ln ((root(5)(x^2+4))/(sqrt(x^2-8)))-ln(x-1)=`

`ln(root(5)(x^2+4)/((x-1)sqrt(x^2-8)))` ** <-- Your solution**

`1/5log(x^2+4) -1/2log(x^2-8)-log(x-1)=`

`=log root(5) (x^2+4)-log sqrt(x^2-8)-log(x-1)=`

`=log(root(5)(x^2+4))/((x-1)sqrt(x^2-8))`