Question

Expanding Logarithmic Expressions In Exercise, use the properties of logarithms to rewrite the expression as the sum, difference, or multiple of logarithms.

In(x - 1/x + 1)^2

Answer 1

We can use two properties to solve:

Quotient Rule:
`\text{ln}(x)/(y) = \text{ln}(x)-\text{ln}(y)`

Power Rule:
`\text{ln}(x)^p = p~\text{ln}(x)`

Simplify the expression using the power rule:

In(x - 1/x + 1)^2 → 2 ln (x - 1/x + 1)

Simplify using the quotient rule:

2 ln (x - 1/x + 1) → 2[ln(x - 1) - ln(x + 1)]

Therefore, the the simplified logarithm is 2[ln(x - 1) - ln(x + 1)]

Best of Luck!

Answer 2

Answer:
`2[\ln(x-1)-\ln(x+1)]`

Explanation:

We know that , the properties of logarithm are:

P[1]
`\log (ab)= \log a+\log b`

P[2]
`\log((a)/(b))=\log a-\log b`

P[3]
`\log a^n= n\log a`

The given expression in terms of Natural log :
`\ln ((x-1)/(x+1))^(2)`

`=2\ln (((x-1)/(x+1))` [ By using P[3]]

`=2[\ln(x-1)-\ln(x+1)]` [ By using Property (1)]

Hence, the simplified expression becomes
`2[\ln(x-1)-\ln(x+1)]` .