To combine the expression 3 ln(x) + 2 ln(c) into a single natural logarithm, we can utilize the properties of logarithms. Specifically, we will be using the power rule and the product rule of logarithms.
- Power Rule: The power rule states that
n ln(a) = ln(a^n). This means you can move the coefficient of the logarithm as the exponent of the argument. - Product Rule: The product rule states that
ln(a) + ln(b) = ln(a * b). This allows us to combine two logarithm terms into one by multiplying their arguments.
Let’s apply these rules step by step:
- First, we apply the power rule to both terms:
3 ln(x) = ln(x^3)2 ln(c) = ln(c^2)
Now our expression looks like this:
ln(x^3) + ln(c^2)
Next, we apply the product rule:
ln(x^3) + ln(c^2) = ln(x^3 * c^2)
So, the final expression as a single natural logarithm is:
Answer: ln(x^3 * c^2)
This means that the combined expression of 3 ln(x) + 2 ln(c) can be neatly summarized as a single natural logarithm: ln(x^3 * c^2).