Question

Using Properties of logarithms In Exercise,Use the properties of logarithms and a fact that In 2≈ 0.6931 and In 3 ≈1.0986 to approximate the logarithm.Then use a calculator to confirm your approximation.

(a) In 6
(b) In 3/2
(c) In 81
(d) In (3)1/2

Answer 1

1) 1.7917

2) 0.4055

3) 4.3944

4) 0.5493

Explanation:

We are given the following:

`\ln 2 \approx 0.6931\\\ln 3 \approx 1.0986`

We have to evaluate the following:

1) In 6

Logarithmic property:

`\ln (a* b) = \ln a  + \ln b`

`\ln 6 = \ln (2* 3)\\= \ln 2 +\ln 3\\= 0.6931 + 1.0986\\=1.7917`

2)
`\ln ((3)/(2))`

Logarithmic property:

`\ln (a)/(b) = \ln a - \ln b`

`\ln (3)/(2) = \ln 3 - \ln 2 \\\\= 1.0986 - 0.6931\\=0.4055`

3) In 81

Logarithmic property:

`\ln(a^x) = x\ln a`

`\ln 81 = \ln(3^4)\\=4\ln 3\\=4* 1.0986\\=4.3944`

4)
`\ln (3)^{(1)/(2)}`

Logarithmic property:

`\ln(a^x) = x\ln a`

`\ln (3)^{(1)/(2)}\\\\=(1)/(2)\ln 3\\\\=(1)/(2)* 1.0986\\\\=0.5493`