log(10^7) = x if and only if 10^x = 10^7 which makes x = 7.
log(10^5) = y if and only if 10^y = 10^5 which makes y = 5.
log(10^35) = z if and only if 10^z = 10^35 which makes z = 35.
since z = 35, and 35 = 5 * 7, and y = 5 and x = 7, then you can say that z = x * y.
therefore, log(10^35) = x * y.
another way you can look at it is as follows:
log(10^7) = x
log(10^5) = y
log(10^35) is equal to log(10^(5*7) which is equal to log((10^5)^7) which is equal to 7 * log(10^5) which is equal to 7 * 5 * log(10) which is equal to 7 * 5 * 1 which is equal to 35.
since you know that x = 5 and y = 7, and you know that 35 = 5 * 7, then you know that this is equivalent to log(10^35) is equal to x * y.