SOLUTION: How di I simplify this equation 2ln6+3ln4-5ln(1/3)

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Question 272536: How di I simplify this equation
2ln6+3ln4-5ln(1/3)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2ln%286%29%2B3ln%284%29-5ln%281%2F3%29
These are not like terms so we cannot add and subtract them. But logarithms have properties which allow you to manipulate them is various ways:
  • log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  • log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29
  • q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29

The first one allows you to combine two logarithms which have a "+" between them and the second one allows you to combine two logarithms which have a "-" between them. These two properties require that the bases of the two logarithms are the same and that the coefficients of the two logarithms are 1's. The third property allows you to move a coefficient from in front into the argument as an exponent.

So to simplify your expression (NOT equation) we will start with cahnging the coefficients to 1's using the third property. This will then allow us to use the first two properties to combine the logarithms. Using the third property on your expression we get:
ln%286%5E2%29%2Bln%284%5E3%29-ln%28%281%2F3%29%5E5%29
which simplifies to:
ln%2836%29%2Bln%2864%29-ln%281%2F243%29
Now we can use the first property to combine the first two logarithms:
ln%2836%2A64%29-ln%281%2F243%29
which simplifies to:
ln%282304%29+-+ln%281%2F243%29
Next we can use second property to combine the remaining two logarithms:
ln%282304%2F%281%2F243%29%29
which simplifies to:
ln%282304%2A243%2F1%29%29
ln%28559872%29