Question 405048
{{{log(5, (8sqrt(y/m)))}}}
There are three properties of logarithms that are used in these "expansion" problems:<ul><li>{{{log(a, (p*q)) = log(a, (p)) + log(a, (q))}}}</li><li>{{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}}</li><li>{{{log(a, (p*q)) = q*log(a, (p))}}}</li></ul>
Your logarithm's argument is the product of 8 and the square root. So we can use the first property, the one for products, to do the first expansion:
{{{log(5, (8)) + log(5, (sqrt(y/m)))}}}
The next step is the trickiest. Remembering that a square root is the same as a power of 1/2 we can rewrite the square root. And by rewriting the argument this way we can use the third property above to expand it:
{{{log(5, (8)) + log(5, ((y/m)^(1/2)))}}}
Using the third property on the second logarithm we get:
{{{log(5, (8)) + (1/2)log(5, (y/m))}}}
Next, since the argument of the second logarithm is now a quotient, we can use the second rule to expand it:
{{{log(5, (8)) + (1/2)(log(5, (y)) - log(5, (m)))}}}
Note the use of parentheses! This is an excellent habit to have when substituting one expression for another, especially when the two expressions have a different number of terms. Here we are replacing a single logarithm with the difference of two logarithms (one term replaced by two terms). Without the parentheses we might not realize that the 1/2 applies to both terms! With the parentheses it should be obvious that the Distributive Property must be used:
{{{log(5, (8)) + (1/2)log(5, (y)) - (1/2)log(5, (m)))}}}
This is the "expanded" expression, with each logarithm's argument being a simgle number or variable.