Question 392349
{{{6log(5, (x)) + log(5, (y))}}}
These are not like terms so we cannot just add them. (If it had been {{{6log(5, (x)) + log(5, (x))}}} then they would have been like terms and adding them would give us {{{7log(5, (x))}}} (just like 6q + q = 7q!)<br>
But there is another way to combine two logarithms with a "+" between them into one. A property of logarithms, {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}, allows us to combine two logarithms if all of the following are true:<ul><li>There is a "+" between them.</li><li>The bases of the two logarithms are the same</li><li>The coefficients of the logarithms are 1's.</li></ul>
Your expression meets the first two requirements. But the first logarithm has a coefficient of 6. If we could eliminate that 6 then we would be in business. Fortunately there is another property of logarithms, {{{q*log(a, (p)) = log(a, (p^q))}}}, which allows us to "move" a coefficient into the argument of the logarithm as an exponent. Using this property on the first term we get:
{{{log(5, (x^6)) + log(5, (y))}}}
Now we can use the first property to combine them into one:
{{{log(5, (x^6*y))}}}