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In this example, we’re asked to expand the given logarithmic expression, log base 3 of M squared N to the 5th.  Remember that our first law of logarithms states that if two values are multiplied together inside a logarithm, such as M squared times N to the 5th, then we can expand the logarithm into the sum of two separate logarithms, in this case log base 3 of M squared plus log base 3 of N to the 5th.  Next, notice that each logarithm has a power inside the logarithm, and remember that our third law of logarithms states that if we have a power inside a logarithm, we can move the exponent to the front of the logarithm, so we have 2 times log base 3 of M + 5 times log base 3 of N.

Logarithm Rules: Expanding Logarithms | MathHelp.com