Question 815816
{{{5log((3))+log((4))}}}
<ul><li>To go from two logs to one we need to either eliminate one of them or find a way to combine them. There's no way to eliminate a log from this expression so we will have to combine them.</li><li>There are two ways to combine logs:<ul><li>Algebraically add them. This requires that the bases and arguments of the logs are the same. The bases of our logs are the same, 10, but the arguments are different, 3 and 4. Since there is no way to make the arguments be the same we will not be able to add the logs algebraically.</li><li>Use the {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}} property of logs. This property requires that the logs have the same bases (which ours do) and have coefficients of 1 (which ours do not since the first log has a coefficient of 5). Fortunately there is a way to change the coefficient to a 1.</li></ul></li><li>A property of logs, {{{n*log(a, (p)) = log(a, (p^n))}}}, allows one to "move" a coefficient into the argument as its exponent. This is how we will handle the 5.</li></ul>So our plan is to move the 5 using one property and then combine the logs using the other:
{{{log((3^5))+log((4))}}}
which simplifies to:
{{{log((243))+log((4))}}}
Now we can use the other property to combine them:
{{{log((243*4))}}}
which simplifies to:
log(972)