Let log (A) = − 8 , log (B) = 1 , and log (C) = 9 . Evaluate the following logarithms using logarithmic properties.

Question

Let log (A) = − 8 , log (B) = 1 , and log (C) = 9 . Evaluate the following logarithms using logarithmic properties.

log(a^5/√b) =

log (b/c^3) =

log (a/b^4c) =

please give answer and explain i'm havin trouble with this topic

Answer 1

Hi Leon,

So, when dealing with logs, you need to remember some information. When you have divisions, you can subtract the log of whatever is on top from the log of whatever is on the bottom like this:

Log₁₀(A/B)=Log₁₀A - Log₁₀B

When you have a multiplication, then instead of subtracting, you have addition of the two:

Log₁₀(AB)=Log₁₀A+Log₁₀B

And, when you have a power, the following applies:

Log₁₀(A^4)= 4 x Log₁₀A

Also, don't forget how logs work:

Log_nm=d means n^d=m

Using the above information, you can find the answer to your question easily. Don't forget that a square root is the exponent of 0.5.

Now, lets take a look at the first question. For the purposes of making typing easier, I am not going to type the 10 in each log. Also, generally speaking when the base of the log is not given, we assume it is 10.

a) Log(a^5/√b) = log(a^5) - log(b^0.5) = 5log(a) - 0.5log(b) = 5(-8) - 0.5(1) = -40 - 0.5 = -40.5

b) Log(b/c³) = log(b) - log(c³) = log(b) - 3log(c) = 1 - 3(9) = 1 - 27 = -26

c) For this one, I was not sure if the c is in the exponent or if it is multiplied by the denominator. I have included the answer for both.

First if

Log(a/(b⁴c)) = log(a) - log(b⁴c) = log(a) - (4log(b) + log(c)) = -8 - (4(1)+9) = -8 - 13= -21

Second if

Log(a/b^4c) = Log(a) - Log(b^4c) = -8 - 4c.Log(b) = -8 - (4 x 10⁹ x 1) = -8 - 4 x 10⁹ and since -4 x 10⁹ is a huge number, we can disregard the -8 and the answer would be -4 x 10⁹

REMEMBER: Log₁₀C=9 which means C=10⁹

I believe the first scenario is what the question is asking, but I wanted to make sure you know how to get to the answer either way.