Question 268185
{{{log(a, (7)) + log(a, (12))}}}<br>
Logarithms can only be added if the bases and the arguments are the same. Your logarithms have the same base, a, but the arguments are different. So we cannot add them.<br>
So how can {{{log(a, (7)) + log(a, (12))}}} be expressed as a single logarithm like all the answers? Well, we have a property of logarithms which allows us to combine two logarithms which have a addition symbol between them: {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}. This property only requires that the bases are the same (and the logarithms have coefficients of 1). Using this on your expression we get:
{{{log(a, (7*12))}}}
which simplifies to
{{{log(a, (84))}}}<br>
When the bases and arguments are the same we can add (or subtract) logarithms. But when we do so, only the coefficients change. After the addition (or subtraction) the bases and arguments of the logarithm <b>are still the same</b>! For example:
{{{log(a, (15)) + 3log(a, (15)) = 4log(a, (15))}}}
It's just like
x + 3x = 4x.