Question 392745
3ln(5)+2ln(x)
These are not like terms so we cannot simply add them. But there is a property, {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}, which can be used to combine two logarithms into one. This property requires all of the following:<ul><li>There is a "+" between the two logarithms. (Note: there is also another property for when there is a "-" between two logarithms.)</li><li>The bases of the two logarithms must be the same.</li><li>The coefficients of the two logarithms must be 1's.</li></ul>
Your expression meets the first two requirements but not the third one. Fortunately there is another property of logarithms, {{{q*log(a, (p)) = log(a, (p^q))}}}, which allows us to move a coefficient of a logarithm into its argument as an exponent. Using this property on both of your logarithms we get:
{{{ln(5^3)*ln(x^2)}}}
which simplifies to:
{{{ln(125)*ln(x^2)}}}
Now that the coefficients are 1's we can use the other property to combine the two logarithms:
{{{ln(125*x^2)}}}