document.write( "Question 392745: Write the expression as a single natural logarithm. 3 ln 5+2 ln x \n" ); document.write( "

Algebra.Com's Answer #279038 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
3ln(5)+2ln(x)
\n" ); document.write( "These are not like terms so we cannot simply add them. But there is a property, $\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"$ , which can be used to combine two logarithms into one. This property requires all of the following:

There is a \"+\" between the two logarithms. (Note: there is also another property for when there is a \"-\" between two logarithms.)
The bases of the two logarithms must be the same.
The coefficients of the two logarithms must be 1's.

\n" ); document.write( "Your expression meets the first two requirements but not the third one. Fortunately there is another property of logarithms, $\"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\"$ , 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:
\n" ); document.write( " $\"ln%285%5E3%29%2Aln%28x%5E2%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"ln%28125%29%2Aln%28x%5E2%29\"$
\n" ); document.write( "Now that the coefficients are 1's we can use the other property to combine the two logarithms:
\n" ); document.write( " $\"ln%28125%2Ax%5E2%29\"$ \n" ); document.write( "

\n" );