Question 718443
{{{5log(2, (x))+(1/2)log(2, (y))}}}
We have two logs which we want to write as one log. There are two ways to combine two logarithmic terms into one:<ul><li>If they are like terms we can add or subtract them (depending on whether there's a "+" or "-" between them. Like logarithmic terms have the same bases and the same argument.</li><li>Use one of the following properties of logarithms:<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></ul>These properties require logs with the same base and coefficients of 1.</li></ul>Our logs have the same bases, 2. But the arguments, x and y, are different. So we cannot add them together. Also our logs have coefficients, 5 and 1/2, which are not 1's. So we cannot use the properties... yet. Fortunately there is another property of logs, {{{log(a, (p^q)) = q*log(a, (p))}}}, which will help. The right side shows a log with a coefficient which is not a 1. The left side shows us how to rewrite the right side with a coefficient of 1: Just move the coefficient into the argument as its exponent!<br>
So we start by using the property which lets us move the coefficients:
{{{log(2, (x^5))+log(2, (y^(1/2)))}}}
Since an exponent of 1/2 means square root, I'm going to rewrite it as a square root:
{{{log(2, (x^5))+log(2, (sqrt(y)))}}}<br>
Now we will use {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}} (because its logs, like ours, have a "+" between them) to combine the logs:
{{{log(2, (x^5*sqrt(y)))}}}