Please condense the expression.
1/2ln(x^2+1)-4ln1/2-1/2[ln(x-4)+lnx]
To condense the given expression, we can utilize the properties of logarithms. Remember that the natural logarithm of the product of two numbers is equal to the sum of the natural logarithm of each number. Additionally, the natural logarithm of a fraction is equal to the natural logarithm of the numerator minus the natural logarithm of the denominator.
Let's simplify step by step:
1/2ln(x^2+1) - 4ln(1/2) - 1/2[ln(x-4) + ln(x)]
First, let's simplify the term 4ln(1/2) using the property mentioned earlier:
4ln(1/2) = ln(1/2^4) = ln(1/16)
Now let's simplify the expression inside the brackets by applying the sum of logarithms property:
1/2[ln(x-4) + ln(x)] = 1/2ln((x-4)(x))
So the condensed form of the expression is:
1/2ln(x^2+1) - ln(1/16) - 1/2ln[(x-4)(x)]