SOLUTION: If log_b 2 = x and log_b 3 = y, evaluate the following in terms of x and y: (a) log_b 432 = (b) log_b 162 =

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: If log_b 2 = x and log_b 3 = y, evaluate the following in terms of x and y: (a) log_b 432 = (b) log_b 162 =      Log On


   

Question 370846: If log_b 2 = x and log_b 3 = y, evaluate the following in terms of x and y:
(a) log_b 432 =
(b) log_b 162 =
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28b%2C+%282%29%29+=+x
log%28b%2C+%283%29%29+=+y
In order to rewrite an expression in terms of x and y, given the above, we will need to find a way to rewrite 432 and 162 in terms of 2's and 3's. Prime factorization will help:
432+=+2%2A2%2A2%2A2%2A3%2A3%2A3+=+2%5E4%2A3%5E3
So log%28b%2C+%28432%29%29+=+log%28b%2C+%282%5E4%2A3%5E3%29%29
Now we can use a property of logarithms, log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29, to split the argument giving us:
log%28b%2C+%282%5E4%29%29+%2B+log%28b%2C+%283%5E3%29%29
Next we can use another property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent of each argument out in front:
4%2Alog%28b%2C+%282%29%29+%2B+3%2Alog%28b%2C+%283%29%29
And last of all we replace the two logarithms with x and y respectively:
4*(x) + 3*(y)
or
4x+3y

Repeating all of the steps above for the other logarithm: