Express the given quantity in a single logarithm.
ln(a + b) + ln(a - b) - 2 ln c
The sum of two logarithms is equal to the logarithm of the product, and the difference equals the quotient:
log(A)+log(B)-log(C)
=log(AB/C)
Twice the logarithm of a quantity is the logarithm of the square of the quantity:
2log(C)=log(C²)
So if we put it all together, we get:
log(A)+log(B)-2log(C)
=log( AB/C²)
To express the given quantity in a single logarithm, we can apply the properties of logarithms. Specifically, we'll use the following properties:
1. Product Rule: log base a (xy) = log base a (x) + log base a (y)
2. Quotient Rule: log base a (x / y) = log base a (x) - log base a (y)
3. Power Rule: log base a (x^y) = y * log base a (x)
Using these properties, we can simplify the given expression step by step.
First, let's apply the product rule to ln(a + b) + ln(a - b):
ln(a + b) + ln(a - b) = ln((a + b)(a - b))
Notice that (a + b)(a - b) simplifies to (a^2 - b^2), so our expression becomes:
ln(a^2 - b^2)
Now, let's substitute this simplified expression back into the original equation:
ln(a^2 - b^2) - 2 ln(c)
Next, we'll apply the power rule to the second term, 2 ln(c):
2 ln(c) = ln(c^2)
Finally, we'll apply the quotient rule to simplify the expression:
ln(a^2 - b^2) - ln(c^2) = ln((a^2 - b^2) / (c^2))
So, the given quantity can be expressed in a single logarithm as ln((a^2 - b^2) / (c^2)).