How do you simplify using a single logarithm: log (A^3)- log (B^(2/3))+ log (A^(1/3)+ log (B^(5/3))

Question

How do you simplify using a single logarithm: log (A³)− log (B^2/3)+ log (A^1/3)+ log (B^5/3)

Answer 1

You will apply the properties of logarithms.

 

Bring any exponents down as the coefficient of the logarithm.

The sum of logs is the product of the log's arguments.

The difference in logs is the quotient of the log's arguments.

 

First, get all the positive log terms with common base arguments.  This way when compacting them, we can apply the rule of exponents of common bases arguments.

 

 

log(A³) + log(A^1/3) - [log(B^2/3) - log(B^5/3)]

 

 

Now, we apply the properties of logarithms.

 

log(A^10/3) - log(1/B) =

 

log(A^9/3 * A^1/3) - log(1/B) =

 

log[(A³ * ³√(A) * B]

Answer 2

log(A³) - log(B^2/3) + log(A^1/3) + log(B^5/3)

 

= log(A³/B^2/3) + log(A^1/3B^5/3)

 

= log[A³A^1/3B^5/3/B^2/3]

 

= log(A^10/3B)

Answer 3

I will help you combine everything into a single logarithm, and then you can simplify the argument of the single logarithm:

 

Use the properties log(A) + log(B) = log(AB), and log(A) - log(B) = log(A/B).

 

log(A^3) - log(B^(2/3)) + log(A^(1/3)) + log(B^(5/3)) = log[A^3 / B^(2/3)] + log[A^(1/3) * B^(5/3)]

 

= log{[A^3 / B^(2/3)] * [A^(1/3) * B^(5/3)]}

 

You can wrap it up from here.