Given that a = log 7 and b = \(\log\) 2, express log 35 in terms of a and b.
Answer Details
We can use the logarithmic identities to simplify the expression for log 35 in terms of a and b.
Firstly, we can write 35 as the product of 7 and 5:
35 = 7 x 5
Next, we can use the logarithmic identity:
log (a x b) = log a + log b
to express log 35 in terms of log 7 and log 5 as follows:
log 35 = log (7 x 5) = log 7 + log 5
Now, we need to express log 5 in terms of a and b.
We can write 5 as the product of 2 and 2.5:
5 = 2 x 2.5
Using the logarithmic identity, we get:
log 5 = log (2 x 2.5) = log 2 + log 2.5
We can express log 2.5 in terms of log 10 (which is equal to 1) and log 2 as follows:
log 2.5 = log (2.5/1) = log (5/2) = log 5 - log 2
Substituting this into our expression for log 35, we get:
log 35 = log 7 + log 5 = log 7 + (log 5 - log 2)
Finally, we can substitute the given values of a and b into the expression above to obtain:
log 35 = a + (log 5 - b)
Simplifying further, we get:
log 35 = a + log 5 - b
Therefore, the answer is (c) a - b + 1.