Final answer:
By applying the properties of logarithms, we can evaluate log 35 - log 6 by finding individual logarithms of prime factors and their differences, resulting in -0.2342. For log 7.5, using the properties and given values, we find that log 7.5 is 1.8751.
Step-by-step explanation:
The question involves evaluating logarithmic expressions using the provided values for log 2, log 5, and log 7. Logarithms have certain properties that can be used to simplify these expressions, one of which is that the logarithm of a product is the sum of the logarithms, and the logarithm of a quotient is the difference of the logarithms.
Evaluation of log 35 - log 6
To evaluate log 35 - log 6:
We can split log 35 into log(5*7) which equals log 5 + log 7.
Similarly, log 6 can be split into log(2*3) which equals log 2 + log 3.
We use the provided values for log 2 (0.3010) and log 5 (0.6989), and calculate log 7 (0.8451), but we still need log 3.
To find log 3, we note that log 3 = log(6/2) = log 6 - log 2. As 6 is between the powers of 10, we estimate log 6 to be roughly halfway between log(10) which is 1 and log(100) which is 2, thus log 6 is around 1.7782.
So, log 3 = log 6 - log 2 = 1.7782 - 0.3010 = 1.4772.
Finally, we subtract log 6 from log 35: (log 5 + log 7) - (log 2 + log 3) = (0.6989 + 0.8451) - (0.3010 + 1.4772) = 1.5440 - 1.7782 = -0.2342.
Evaluation of log 7.5
To evaluate log 7.5, we split it into log(7.5) = log(15/2) = log 15 - log 2.
Find log 15 by splitting it into log(3*5) = log 3 + log 5.
So, log 15 = log 3 + log 5 = 1.4772 + 0.6989 = 2.1761.
Then, subtract log 2: log 15 - log 2 = 2.1761 - 0.3010 = 1.8751.
Therefore, log 7.5 = 1.8751.
Evaluation of log 1500 and log 0.72
Unfortunately, the question has some typos or missing information, which makes it unclear how the second part should be evaluated. Without a clear instruction, we cannot proceed with this part.