
log _{x} a^{m} = m log _{x} a. Proof: Let log _{x} a^{m }= p then x^{p} = a^{m}I log _{x} a = q then a = x^{q} II From I and II, a^{m} = (x^{q})^{m} = x^{qm; }[Since (x^{m})^{n} = x^{mn}]III From I and III, we get a^{m} = a^{p} = x^{qm} Therefore, p = qm Therefore, log _{x} a^{m} = m log _{x} a The logarithm of any power of a number is equal to the product of the logarithm of the number and the index of the power.
Example: Directions: Solve the following problems. Also write at least ten examples of your own. 