Questions: Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. Assume that the variables represent positive real numbers. log6(a^6 / bc^3) = □

Solution Steps

The logarithm of a quotient is the difference of the logarithms. Therefore: \[ \log _{6}\left(\frac{a^{6}}{b c^{3}}\right) = \log _{6}(a^{6}) - \log _{6}(b c^{3}) \]

The logarithm of a power can be rewritten as the exponent times the logarithm. Thus: \[ \log _{6}(a^{6}) = 6 \log _{6}(a) \]

The logarithm of a product is the sum of the logarithms. Therefore: \[ \log _{6}(b c^{3}) = \log _{6}(b) + \log _{6}(c^{3}) \]

Rewrite \(\log _{6}(c^{3})\) using the power rule: \[ \log _{6}(c^{3}) = 3 \log _{6}(c) \]

Substitute the results from Steps 2, 3, and 4 back into the expression: \[ \log _{6}\left(\frac{a^{6}}{b c^{3}}\right) = 6 \log _{6}(a) - \left(\log _{6}(b) + 3 \log _{6}(c)\right) \]

Distribute the negative sign and simplify: \[ \log _{6}\left(\frac{a^{6}}{b c^{3}}\right) = 6 \log _{6}(a) - \log _{6}(b) - 3 \log _{6}(c) \]

Final Answer