Question 370846
{{{log(b, (2)) = x}}}
{{{log(b, (3)) = 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*2*2*2*3*3*3 = 2^4*3^3}}}
So {{{log(b, (432)) = log(b, (2^4*3^3))}}}
Now we can use a property of logarithms, {{{log(a, (p*q)) = log(a, (p)) + log(a, (q))}}}, to split the argument giving us:
{{{log(b, (2^4)) + log(b, (3^3))}}}
Next we can use another property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, to move the exponent of each argument out in front:
{{{4*log(b, (2)) + 3*log(b, (3))}}}
And last of all we replace the two logarithms with x and y respectively:
4*(x) + 3*(y)
or
4x+3y<br>
Repeating all of the steps above for the other logarithm:
{{{log(b, (162)) = log(b, (2*3*3*3*3)) = log(b, (2*3^4)) = log(b, (2)) + log(b, (3^4)) = log(b, (2)) + 4*log(b, (3)) = (x) + 4*(y) = x + 4y}}}