SOLUTION: Write the expression as a single natural logarithm. 3 ln 5+2 ln x

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Question 392745: Write the expression as a single natural logarithm. 3 ln 5+2 ln x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3ln(5)+2ln(x)
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.

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:
ln%285%5E3%29%2Aln%28x%5E2%29
which simplifies to:
ln%28125%29%2Aln%28x%5E2%29
Now that the coefficients are 1's we can use the other property to combine the two logarithms:
ln%28125%2Ax%5E2%29