Combine to one term: 3[ln(x)-2ln(x^2+1)]+2ln5

Question

It is from the book Precalculus with Limits- A Graphing Approach (3rd Edition,) and is on page 277. It is question 86

Answer 1

We use the formula 

1. ln(ab) = ln(a ) +ln(b)

2. ln(a/b) = ln(a) - ln(b)

3. a*ln(b) = ln(b^a)

 

Given expression 

= 3[ln(x) -2ln(x^2+1)] +2ln5

 

= 3[ ln(x) - ln(x^2 +1)^2] + ln(5^2)

 

= 3[ ln ( x / {x^2+1}^2] + ln25

 

= ln [ x / {x^2+1}^2]^3 + ln25

 

= ln [(x^3) / (x^2 +1)^6 ] +ln25

 

= ln 25 * [(x^3) / (x^2 +1)^6 ] 

 

Note : I dont have the text to ref to the exact problem , but hopefully this simplification is useful.