Question 812884
{{{log(b, ((x^3 * y^2 * z^4)^(1/4)))}}}
There are three properties of logarithms that are often used in problems like this:<ul><li>{{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}</li><li>{{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}</li><li>{{{log(a, (p^n)) = n*log(a, (p))}}}</li></ul>Since the argument of our log contains just products and powers we will be using just the first and last properties.<br>
First we will use the last property which allows us to move the exponent of the argument out in front:
{{{(1/4)*log(b, (x^3 * y^2 * z^4))}}}
The argument is now a product. So we will use the first property to split the log of the product into the sum of the logs of the factors:
{{{(1/4)*(log(b, (x^3)) + log(b, (y^2)) + log(b, (z^4)))}}}
Note the parentheses around the logs. It is a good habit to put parentheses around a substitution, especially if there are more terms. The arguments of the three logs are all powers. So we will use the last property again to move the exponents out in front:
{{{(1/4)*(3*log(b, (x)) + 2*log(b, (y)) + 4*log(b, (z)))}}}<br>
Since the expression is now in terms of logs of x, y and z we might be done. But we should probably use the Distributive Property to multiply out the 1/4:
{{{(3/4)*log(b, (x)) + (1/2)*log(b, (y)) + log(b, (z))}}}<br>