Find the value of \( \log_3 250 - \frac{1}{3} \log_3 8 \). - Scottish Highers Maths - Question 6 - 2018

To solve this expression, we will use the properties of logarithms. First, we can apply the power rule of logarithms: ( a \log_b c = \log_b (c^a) ).

This gives us:

[ \log_3 250 - \log_3 (8^{1/3}) ]

Next, according to the quotient rule, which states that ( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) ), we can rewrite the expression:

[ \log_3 \left( \frac{250}{8^{1/3}} \right) ]

Next, we can calculate ( 8^{1/3} ):

[ 8^{1/3} = 2 ]

Now substituting it back in:

[ \log_3 \left( \frac{250}{2} \right) = \log_3 (125) ]

Finally, we know that ( 125 = 5^3 ), so we rewrite the logarithm:

[ \log_3 (125) = \log_3 (5^3) ]

Applying the power rule one more time:

[ = 3 \log_3 (5) ]