Question

I have a log problem that deals with properties of logarithms picture included

Answer 1

The Properties of Logarithms are shown below:

• a)

To rewrite the first logarithm, we have to remember that:

`(3)/(4)=0.75`

Therefore, we can rewrite the expression as follows:

`\log _a(0.75)=\log _a((3)/(4))`

Using the property of division of the logarithms we get:

`=\log _a((3)/(4))=\log _a(3)-\log _a(4)`

Replacing the given values:

`=0.62-0.78=-0.16`

• b)

Also, if we multiply 3 times 4 we get 12. Thus, we can rewrite the second expression:

`\log _a(12)=\log _a(3*4)`

Using the multiplication property of the logarithm:

`=\log _a(3*4)=\log _a(3)+\log _a(4)`

Replacing the values:

`=0.62+0.78=1.4`

• c)

Finally, for the last expression we have to remember that a square root can also be written as an exponent:

`\log _a(\sqrt[]{3})=\log _a(3^{(1)/(2)})`

Then, using the exponentiation property of the logarithms we can rewrite that last expression:

`=\log _a(3^{(1)/(2)})=(1)/(2)\log _a(3)`

As we already know the value of loga(3), we can just replace it and get the result:

`=(1)/(2)\cdot0.62=0.31`

Answer:

• a) -0.16

,

• b) 1.4

,

• c) 0.31