Given that x=logbase3 5 and
y=logbase3 2 , rewrite logbase3 60 in terms of x and y.
To rewrite log base 3 of 60 in terms of x and y, we need to express 60 as a product or quotient of numbers that can be written in terms of x and y.
Let's first consider the prime factorization of 60: 60 = 2^2 * 3 * 5.
Now, we can express 60 in terms of x and y as follows:
log base 3 of 60 = log base 3 of (2^2 * 3 * 5)
Using the properties of logarithms, we can break down this expression:
log base 3 of (2^2 * 3 * 5) = log base 3 of 2^2 + log base 3 of 3 + log base 3 of 5
Since x = log base 3 of 5 and y = log base 3 of 2, we can substitute these values:
log base 3 of 60 = (2 * log base 3 of 2) + log base 3 of 3 + x
Remember that log base 3 of 3 is equal to 1, so we can simplify further:
log base 3 of 60 = 2y + 1 + x
Therefore, log base 3 of 60 can be written in terms of x and y as 2y + 1 + x.
given:
x = log3 5 ----> 3^x = 5
y = log3 2 ----> 3^y = 2
let log3 60 = z ---- 3^z = 60
3^z = 60
= (3)(2)(2)(5)
= (3^0)(3^y)(3^y)(3^x)
= 3(1+y+y+x)
= 3^(x+2y+1)
then z = x+2y+1
log360 = x+2y+1