how to write each expression into a single logarithm.
ln(x-1/x) + ln(x/x+1) - ln((x^(2)-1))
the answer is: -2ln(x+1)
steps please
i tried solving this but ended up getting this long expression
i tried it again and solved it
To write the expression ln(x-1/x) + ln(x/x+1) - ln((x^(2)-1)) into a single logarithm, we can use the properties of logarithms and combine the terms. Here are the step-by-step instructions:
Step 1: Start with the given expression: ln(x-1/x) + ln(x/x+1) - ln((x^(2)-1))
Step 2: Apply the product rule of logarithms to the first two terms:
ln(x-1/x) + ln(x/x+1) = ln((x-1/x) * (x/x+1))
Step 3: Simplify the expression inside the first logarithm:
(x-1/x) * (x/x+1) = ((x-1) * (x)) / (x * (x+1))
Step 4: Cancel out common factors:
((x-1) * (x)) / (x * (x+1)) = (x-1) / (x+1)
Step 5: Apply the quotient rule of logarithms to the simplified expression:
ln((x-1) / (x+1)) - ln((x^2-1))
Step 6: Apply the quotient rule of logarithms to the remaining terms:
ln((x-1) / (x+1)) - ln((x+1) * (x-1))
Step 7: Simplify the expression inside the second logarithm:
(x+1) * (x-1) = x^2 - 1
Step 8: Replace the simplified expression:
ln((x-1) / (x+1)) - ln(x^2 - 1)
Step 9: Apply the difference rule of logarithms to the final expression:
ln((x-1) / (x+1)) - ln(x^2 - 1) = ln(((x-1)/(x+1)) / (x^2 - 1))
Step 10: Simplify the expression inside the logarithm using the properties of fractions:
ln(((x-1)/(x+1)) / (x^2 - 1)) = ln((x-1) / ((x+1) * (x-1)))
Step 11: Cancel out the common factor:
ln((x-1) / ((x+1) * (x-1))) = ln(1 / (x+1))
Step 12: Since 1 divided by any number is the same as the number divided by 1, we can simplify further:
ln(1 / (x+1)) = ln(1) - ln(x+1)
Step 13: ln(1) is equal to 0, so we have:
ln(1) - ln(x+1) = 0 - ln(x+1)
Step 14: Simplify the expression:
0 - ln(x+1) = -ln(x+1)
Therefore, the expression ln(x-1/x) + ln(x/x+1) - ln((x^(2)-1)) can be written as -ln(x+1).
To write the given expression as a single logarithm, we can use the properties of logarithms. Specifically, we can combine the three logarithms into one by applying the multiplication and division properties.
1. Start with the given expression:
ln(x-1/x) + ln(x/x+1) - ln(x^(2)-1)
2. Apply the multiplication property of logarithms to combine the first two logarithms:
ln((x-1/x) * (x/x+1)) - ln(x^(2)-1)
Simplifying the numerator further:
ln((x^2-x) / (x(x+1))) - ln(x^(2)-1)
3. Apply the division property of logarithms to create a single logarithm:
ln((x^2-x) / (x(x+1))) / ln(x^(2)-1)
4. Now, we can use the properties of exponents to simplify the expression further. Recall that ln(a) - ln(b) = ln(a/b).
ln((x^2-x) / (x(x+1))) / ln(x^(2)-1)
= ln((x^2-x) / (x(x+1)) * (1 / (x^2-1)))
Simplifying the numerator further:
= ln(1 / (x+1))
5. Finally, we can use the property ln(1/a) = -ln(a) to simplify the expression:
= -ln(x+1)
Therefore, the given expression can be written as -ln(x+1), or equivalently, -2ln(x+1) after applying the exponentiation property ln(a^n) = n * ln(a).