If log 3=a, log 5=b and log 7=c, express log_{15} 7 in terms of a,b and c.
3 = 10^a
5 = 10^b
7 = 10^c
15 = 10^(a+b)
15^(log15 7) = 7 = [10^(a+b)]^log15 7
10^c = 10^(a+b)log15 7
so
c = (a+b) log15 7
log15 7 = c/(a+b)
To express log_{15} 7 in terms of a, b, and c, we can use logarithmic properties and manipulate the given information.
The first step is to express 15 in terms of a, b, and c. We can do this by using the fact that 15 can be written as the product of its prime factors: 15 = 3 * 5.
Now, applying the logarithmic property log(xy) = log(x) + log(y), we have:
log_{15} 7 = log_{(3*5)} 7 = log_3 7 + log_5 7
Next, we substitute the given values of a, b, and c:
log_{15} 7 = a + b + log_7 7
Since log_7 7 equals 1 (logarithm with the same base and argument cancels out to 1), we simplify further:
log_{15} 7 = a + b + 1
Therefore, log_{15} 7 can be expressed as a + b + 1 in terms of a, b, and c.