Questions: Which property of logarithms does this equation demonstrate? Assume all expressions exist and are well-defined. loga 7x^7 + loga x = loga 7x^8 y Power Property Product Property Change of Base Formula Quotient Property

Solution Steps

To determine which property of logarithms is demonstrated by the given equation, we need to analyze the structure of the equation and compare it with known logarithmic properties. The equation involves a sum of logarithms on the left side and a single logarithm on the right side. This suggests the use of the Product Property of logarithms, which states that the sum of logarithms is equal to the logarithm of the product of the arguments.

The left side of the equation is given by: \[ \log_a (7x^7) + \log_a (x) \] Using the change of base formula, we can express this as: \[ \frac{\log(7x^7)}{\log(a)} + \frac{\log(x)}{\log(a)} = \frac{\log(7x^7) + \log(x)}{\log(a)} \] This simplifies to: \[ \frac{\log(7) + \log(x^7) + \log(x)}{\log(a)} = \frac{\log(7) + 7\log(x) + \log(x)}{\log(a)} = \frac{\log(7) + 8\log(x)}{\log(a)} \]

The right side of the equation is: \[ \log_a (7x^8y) \] Using the change of base formula, this can be expressed as: \[ \frac{\log(7x^8y)}{\log(a)} = \frac{\log(7) + \log(x^8) + \log(y)}{\log(a)} = \frac{\log(7) + 8\log(x) + \log(y)}{\log(a)} \]

Now we compare the simplified left side: \[ \frac{\log(7) + 8\log(x)}{\log(a)} \] with the right side: \[ \frac{\log(7) + 8\log(x) + \log(y)}{\log(a)} \] We can see that the left side does not include the term \(\log(y)\). Therefore, the two sides are not equal, which indicates that the equation does not hold true as presented.

Final Answer