Question 409554
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{{{log((12))+(1/2)log((7)) - log((2))}}}
The properties that allow us to combine logarithms which have different arguments, {{{log(a, (p)) + log(a, (q)) = log(a, p*q))}}} and {{{log(a, (p)) - log(a, (q)) = log(a, p/q))}}}, require that the coefficients of the logarithms be 1's. So we need to eliminate the (1/2) in front of the middle log. Fortunately there is another property, {{{q*log(a, (p)) = log(a, (p^q))}}}, which allows us to move a coefficient into the argument as the exponent. So we start by using this third property to move the 1/2:
{{{log((12))+log((7^(1/2))) - log((2))}}}
Since an exponent of 1/2 means square root, I am going to rewrite the argument as a square root:
{{{log((12))+log((sqrt(7))) - log((2))}}}
Now we can start combining. The first two logs have a "+" between them so we will use the first property to combine them:
{{{log((12*sqrt(7))) - log((2))}}}
The remaining logs have a "-" between them so we will use the second property to combine them:
{{{log((12*sqrt(7)/2))}}}
And finally the fraction will reduce:
{{{log((6*sqrt(7)))}}}