write the expression as a single logarithm
In(x/x-2)+In(x+2/x)-In(x^2-4)
Please show work
Basic rules: lna+lnb=ln(ab) and lna-lnb=ln(a/b) also lna^x=xlna
ln[x/(x-2)*(x+2/x)]-ln(x^2-4)= ln[(x^2+2)/(x-2)]-ln(x^2-4)= ln [divide]hence simplify fraction and get answer
To write the given expression as a single logarithm, we can use the properties of logarithms to simplify it step-by-step. Let's break it down:
1. In(x/x-2) + In(x+2/x) - In(x^2-4)
We can first simplify each logarithm individually using the properties of logarithms:
2. ln(x/(x-2)) + ln((x+2)/x) - ln(x^2-4)
Using the property ln(a) + ln(b) = ln(a * b), we can combine the first two logarithms:
3. ln((x/(x-2)) * ((x+2)/x)) - ln(x^2-4)
4. ln((x(x+2))/((x-2)x)) - ln(x^2-4)
The terms (x-2) and (x+2) in the numerator and denominator of the logarithm can be canceled out:
5. ln((x+2)/(x-2)) - ln(x^2-4)
Now, using the property ln(a) - ln(b) = ln(a/b), we can simplify the expression further:
6. ln((x+2)/(x-2)) - ln((x-2)(x+2))
Finally, we can use the property ln(a * b) = ln(a) + ln(b) to combine the two logarithms:
7. ln((x+2)/(x-2)(x+2))
Hence, the given expression can be written as a single logarithm: ln((x+2)/(x-2)(x+2)).