To express 3 ln(3 ln(9)) as a single natural logarithm, we’ll start by simplifying the argument of the logarithm using properties of logarithms.
1. **Rewrite the inner logarithm:** First, let’s deal with ln(9). We can express 9 as a power of 3:
9 = 32
Therefore, we can rewrite this as:
ln(9) = ln(32) = 2 ln(3)
2. **Substitute back into the original expression:** Now, we substitute ln(9) back into our original expression:
3 ln(3 ln(9)) = 3 ln(3 * (2 ln(3)))
3. **Simplify the product inside the logarithm:** This gives us:
3 ln(6 ln(3))
4. **Apply the logarithm property:** Now we can use the logarithmic identity:
a * ln(b) = ln(ba)
So, applying this property:
3 ln(6 ln(3)) = ln((6 ln(3))3)
Finally, we arrive at the expression:
3 ln(3 ln(9)) = ln((6 ln(3))3)
In conclusion, 3 ln(3 ln(9)) can be expressed as a single natural logarithm: ln((6 ln(3))3).