Let's go through each of your questions one by one:
-
Using the product rule to rewrite \( \log (xy) \): The product rule states that the logarithm of a product is the sum of the logarithms: \[ \log (xy) = \log (x) + \log (y) \] So, the answer is: log (x) + log (y)
-
Using the quotient rule to rewrite \( \log (x/y) \): The quotient rule states that the logarithm of a quotient is the difference of the logarithms: \[ \log (x/y) = \log (x) - \log (y) \] So, the answer is: log (x) - log (y)
-
Using the power rule to rewrite \( \log (8) \): The power rule states that the logarithm of a power is the exponent times the logarithm of the base: Since \( 8 = 2^3 \), we have: \[ \log (8) = \log (2^3) = 3 \log (2) \] So, the answer is: 3 log (2)
-
Identifying the original base from the captionless image: Since I cannot see the image, I cannot directly identify the base for you. However, if the context is related to logarithms and the options provided are "3", "27", and "243", it is likely that these are powers of 3:
- \( 3 = 3^1 \)
- \( 27 = 3^3 \)
- \( 243 = 3^5 \)
If you can provide the context or any additional information regarding the image, I can help you identify the base more accurately. Otherwise, it is reasonable to say that since \( 27 = 3^3 \) and \( 243 = 3^5 \), they would be the stronger candidates for an original base in a logarithmic context if that's the direction we are headed.
If you have additional details or questions, feel free to ask!