Question 627466
{{{3(ln(x) - 2ln(x^3 + 2)) + 4ln(5)}}}
First of all, these are not "eye-en's", they are "ell-en's". "ell" for logarithm and "en" for natural. These are called natural logarithms.<br>
There are two ways to combine logarithmic terms:<ul><li>Addition or subtraction if they are like terms. (Like logarithmic terms have the same bases and same arguments.)</li><li>Use one of the following properties:<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 bases be the same and the coefficients are 1's.</li></ul>Your logs all have the same base, e. But their arguments are different so we cannot use addition or subtraction to combine the terms.<br>
The coefficients are not 1's either. So it seems that we could not use the properties to combine the terms either. But 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 as its exponent. So we can use this property to make the terms suitable for the other two properties:
{{{3(ln(x) - ln((x^3 + 2)^2)) + ln(5^4)}}}
Since {{{5^4 = 625}}} this simplifies to:
{{{3(ln(x) - ln((x^3 + 2)^2)) + ln(625)}}}
We can now use the second property to combine the logs in the parentheses. (We use the second property because it has a "minus" between the terms, just like our logs.) Using the second property we get:
{{{3(ln(x/(x^3 + 2)^2)) + ln(625)}}}
Before we try to combine the remaining logs we must first use the third property to move the 3 out of the way:
{{{ln((x/(x^3 + 2)^2)^3) + ln(625)}}}
Now we can use the first property (because of the "plus" between the terms) to combine the remaining terms:
{{{ln(((x/(x^3 + 2)^2)^3)*625))}}}
or:
{{{ln(625(x/(x^3 + 2)^2)^3)}}}