Question 258258
{{{2(ln(x-3)+ln(x))-ln(x^(2)-9)}}}
To combine logarithms we use two of the 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 that the coefficients of the logarithms are 1's. For logarithms which have other coefficients, we have a third property which allows us to move a coefficient of a logarithm into the argument as an exponent: {{{q*log(a, (p)) = log(a, (p^q))}}}
Inside the parentheses we find {{{ln(x-3) + ln(x)}}}. Since this is an addition, we can use the first property to combine these:
{{{2(ln((x-3)*(x)))-ln(x^(2)-9)}}}
Next, we can use the third property to move the 2 from in front into the argument of the logarithm:
{{{ln(((x-3)*(x))^2)-ln(x^(2)-9)}}}
And finally, since this is a subtraction, we can use the second property to combine the remaining logarithms:
{{{ln((((x-3)*(x))^2)/(x^(2)-9))}}}
We have now condensed the expression into a single logarithm. This may be an acceptable answer. But we can simplify the argument of the logarithm. The fraction will reduce. After we factor the denominator we get:
{{{ln(((x-3)*(x))^2)/((x+3)(x-3))}}}
We can see that the x-3 in the denominator will cancel with one of the two (x-3)'s (it is squared, after all) leaving:
{{{ln(((x-3)*x^2)/(x+3))}}}
Last of all we can multiply out the numerator:
{{{ln((x^3-3x^2)/(x+3))}}}