To express 3 ln 3 ln 9 as a single natural logarithm, we can start by simplifying the expression step by step.
We know from logarithmic properties that:
- ln(a^b) = b * ln(a)
- ln(a * b) = ln(a) + ln(b)
First, let’s recognize that ln 9 can be rewritten using the property of logarithms:
ln 9 = ln(3^2) = 2 ln 3
Now, substituting this back into the original expression gives us:
3 ln 3 ln 9 = 3 ln 3 (2 ln 3)
Next, we can combine these to simplify:
3 ln 3 (2 ln 3) = 6 (ln 3)^2
Now, we want to express 6 (ln 3)^2 as a single natural logarithm. To do this, we will use the logarithmic property that links multiplication and powers:
6 (ln 3)^2 can be rewritten as:
ln(3^6)
Thus, our final expression for 3 ln 3 ln 9 in terms of a single logarithm is:
ln(729). This is because 3^6 = 729.
Therefore, the expression 3 ln 3 ln 9 expressed as a single natural logarithm is ln(729).