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:

$\frac{3}{4}=0.75$

Therefore, we can rewrite the expression as follows:

$\log _a(0.75)=\log _a(\frac{3}{4})$

Using the property of division of the logarithms we get:

$=\log _a(\frac{3}{4})=\log _a(3)-\log _a(4)$

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:

$\log _a(12)=\log _a(3\times4)$

Using the multiplication property of the logarithm:

$=\log _a(3\times4)=\log _a(3)+\log _a(4)$

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:

$\log _a(\sqrt[]{3})=\log _a(3^{\frac{1}{2}})$

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

$=\log _a(3^{\frac{1}{2}})=\frac{1}{2}\log _a(3)$

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

$=\frac{1}{2}\cdot0.62=0.31$

Answer:

• a) -0.16

,

• b) 1.4

,

• c) 0.31