Use exponent laws to explain why log(40) - log(5) is equivalent to log(8).?

log 40 - log 5 = log (8 • 5) - log 5

                      = log 8 + log 5 - log 5

                      = log 8  .....................ANS

= Log(40) - Log(5)

= Log(5 * 8) - Log(5) → you know that: Log(a * b) = Log(a) + Log(b)

= Log(5) + Log(8) - Log(5)

= Log(8)

= Log(2³) → you know that: Log[x^(a)] = a.Log(x)

= 3.Log(2)

log(40) - log(5) ≟ log(8)

You want to test if this is true using exponent laws.

If we start out by getting the prime factorization of the 40 and 8, we get:

log(40) - log(5) ≟ log(8)

log(2³ * 5) - log(5) ≟ log(2³)

The log of a product is the same as the sum of two logs, so:

log(2³) + log(5) - log(5) ≟ log(2³)

We can pull the exponent out of the logs so the 3 moves outside of each:

3 log(2) + log(5) - log(5) ≟ 3 log(2)

We can now see that log(5) - log(5) is zero because that's true for any real number subtracting itself:

3 log(2) ≟ 3 log(2)

TRUE

Now we have the same expression on both sides of this equation, so this, and the original equation, is true.

log(40) - log(5) = log(8)

You should be familiar with several logarithmic rules by now. If not, look at the image below and put these in your notes.

In particular, you want to use the quotient rule:

log(x/y) = log(x) - log(y)

Use that in reverse:

log(40) - log(5) = log(40/5) 

= log(8)