How do I write the expression 3 ln x + 2 ln c as a single natural logarithm?

To combine the expression 3 ln x + 2 ln c into a single natural logarithm, we can use the properties of logarithms.

The key properties we will use are:

• n ln a = ln aⁿ – This means we can rewrite a coefficient in front of the logarithm as an exponent.
• ln a + ln b = ln(ab) – This allows us to combine two logarithms by multiplying their arguments.

Applying these properties step by step:

1. First, rewrite 3 ln x as ln(x³):
• 3 ln x = ln(x³)

2. Next, rewrite 2 ln c as ln(c²):
• 2 ln c = ln(c²)

Now the expression becomes:

ln(x³) + ln(c²)

Using the second property to combine these, we have:

ln(x³) + ln(c²) = ln(x³ * c²)

Thus, the expression 3 ln x + 2 ln c can be written as a single natural logarithm:

ln(x³ * c²)