Question

Condensing Logarithmic Expression In Exercise,use the properties of logarithms to rewrite the expression as the logarithm of a single quantity.See example 4.

3 In x + 2 In y - 4 In z

Answer 1

We can use three rules to rewrite the logarithm:

Power rule:
`\text{ln}(x^p)=p~\text{ln}(x)`

Product rule:
`\text{ln}(xy)=\text{ln}(x)+\text{ln}(y)`

Quotient rule:
`\text{ln}(x)/(y) = \text{ln}(x)-\text{ln}(y)`

Rewrite each part of the expression:

3 In x → ln(x^3)

2 ln y → ln(y^2)

4 ln z → ln(z^4)

According to the product rule, x^3 and y^2 get multiplied. According to the quotient rule, z^4 is being divided.

ln(x^3) + ln(y^2) - ln(z^4) → x^3*y^2/z^4

Therefore, the logarithm as a single quantity is
`\text{ln}(x^3y^2)/(z^4)`

Best of Luck!

Answer 2

`ln ((x^3y^2))/(z^4)`

Explanation:

`3 ln x + 2 ln y - 4 ln z`

m ln(x)= ln x^m

`3 ln x + 2 ln y - 4 ln z`

`ln x^3 +ln y^2 - ln z^4`

ln(mn)= ln m +ln n

`(ln x^3 +ln y^2)- ln z^4`

`(ln(x^3y^2)-ln z^4`

ln (m)-ln(n)= ln(m/n)

`ln ((x^3y^2))/(z^4)`