If log_8 (60) - log_8 (3) - log_8 (5) is to be simplified using the properties of logarithms, what would be the resultant expression?

Steps to Solve

1. Identify the logarithmic property to use

We will use the property of logarithms that states $ \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right)$.

2. Apply the property to the first two logarithms

Using the property, we can simplify the first part of the expression:

$$ \log_8(60) - \log_8(3) = \log_8\left(\frac{60}{3}\right) $$

3. Calculate the quotient

We calculate the quotient:

$$ \frac{60}{3} = 20 $$

So, we have:

$$ \log_8(60) - \log_8(3) = \log_8(20) $$

4. Combine with the remaining logarithm

Now we apply the logarithmic property again with the next logarithm $\log_8(5)$:

$$ \log_8(20) - \log_8(5) = \log_8\left(\frac{20}{5}\right) $$

5. Calculate the final quotient

Next, we calculate:

$$ \frac{20}{5} = 4 $$

Thus, we have:

$$ \log_8(20) - \log_8(5) = \log_8(4) $$

6. Final simplification

We now recognize that $4 = 8^{2/3}$, so we can express the final result as:

$$ \log_8(4) = \log_8(8^{2/3}) = \frac{2}{3} $$