write as a single logarithm using properties of logarithms.

Question

log5-(2loga+3logb)

Answer 1

Vicky--

 

First, let's recap the three basic properties of logarithms.  Assuming all have the same base (here we are assuming base 10), we have

 

1)  log(a*b) = log(a) + log(b)

2)  log(a/b) = log(a) - log(b)

3)  log(a^r) = r*log(a)

 

So, we can use these three properties to rewrite your expression.  Observe,

 

log(5) - (2*log(a) + 3*log(b)) = log(5) - (log(a^2) + log(b^3))      [by prop. 3]

              =  log(5) - (log(a^2*b^3)     [by prop. 1]

              =  log[5/(a^2*b^3)]            [by prop. 2]

Answer 2

Hello, thank you for taking the time to post your question!

On this type of question you want to apply the properties of logarithms in order to simplify! So first we can use the power rule for the terms inside of the parentheses:

log 5 – (log a^2 + log b^3)

next then I would use the Product Rule for the part inside of the parentheses

log 5 – (log(a^2 * b^3))

finally then you can use the Quotient Rule to combine all the way

log(5 / (a^2 * b^3))

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)