Question

Use properties of logarithms to condense the logarithmic expression below. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.

6 ln x + 3 ln y -2 ln z

Answer 1

To condense the expression 6 ln x + 3 ln y - 2 ln z into a single logarithm, we rewrite it as ln x^6 + ln y^3 - ln z^-2 and then combine them using logarithm properties to get ln(x^6y^3/z^2).

Step-by-step explanation:

To condense the logarithmic expression 6 ln x + 3 ln y - 2 ln z, we need to use the properties of logarithms.

Firstly, we apply the property that expresses the logarithm of a number raised to an exponent: The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. Using this property, we can write:

• 6 ln x as ln x^6
• 3 ln y as ln y^3
• -2 ln z as ln z^-2

Secondly, we apply the properties regarding the logarithm of a product and division:

• The logarithm of a product of two numbers is the sum of the logarithms: ln xy = ln x + ln y
• The logarithm resulting from the division of two numbers is the difference between the logarithms: ln(x/y) = ln x - ln y

By combining these properties, the expression can be condensed into a single logarithm:

ln(x^6y^3/z^2)