When evaluating logarithmic equations, the logarithm power rule can be a useful tool. The logarithmic power rule can also be used to access exponential terms. When a logarithmic term has an exponent, the logarithm power rule says that we can transfer the exponent to the front of the logarithm. Along with the product rule and the quotient rule, the logarithm power rule can be used for expanding and condensing logarithms.
So what we have here is log base 7 of x cubed okay? So knowing that we know about powers, this is just the same thing as log base 7 of x times x times x and using the product rule of multiple, of logarithms what we could do is really break this up so say we have we compare these two logs together and we have this log separately so so what the product rule is saying is we could rewrite this product as a sum ending with the log base 7 of x plus log base 7 of x times x. Going through the same process of splitting this one up over here what we'll end up with then is log base 7 of x plus log base 7 of x plus yet one more log base 7 of x.
What you remember about just combining terms as if we ever have like terms you combine them, so we have 3 of the same things and this will give us 3log base 7 of x okay. This introduces a new property of logs which is the power rule okay? If we ever have a power and whatever is inside the log what we can do is bring that down in front, log base b of x, so this n that exponent can just come down to front and we have the same expression.
One thing to be very careful although, is this is not the same thing as if we have log base b of x to the n okay? This only works when an exponent is on the term inside the log okay? This one is not equal to n log base b of x okay? You can only do it if the exponent is the only thing inside not if the whole logarithm is to the power, so power rule of logarithm if you ever have something in the log that has the power bring the number down in front.