use the properties of logarithms to condense the logarithmic expression. write the expression as a single logiarithm whose coefficient is 1.
2IN(x+3)-3INx
I assume you meant ln
2 ln(x+3) - 3 lnx
= ln(x+3)^2 - ln x^3
= ln [ (x+3)^2/x^3 ]
To condense the logarithmic expression, we can use the properties of logarithms. The properties that will help us in this case are:
1. Product Rule: log base b of (MN) = log base b of M + log base b of N
2. Quotient Rule: log base b of (M/N) = log base b of M - log base b of N
3. Power Rule: log base b of (M^p) = p * log base b of M
Let's simplify the given expression:
2ln(x+3) - 3ln(x)
Using the Product Rule, we can rewrite it as:
ln((x+3)^2) - ln(x^3)
Using the Quotient Rule, we can rewrite it as:
ln((x+3)^2 / x^3)
Therefore, the condensed logarithmic expression is:
ln((x+3)^2 / x^3)
To condense the logarithmic expression using the properties of logarithms and write it as a single logarithm with a coefficient of 1, we can apply the following steps:
1. Start with the given expression:
2ln(x + 3) - 3ln(x)
2. Use the property of logarithms that allows us to combine logarithms with the same base when they are being subtracted:
ln((x + 3)^2 / x^3)
3. Apply the property of logarithms that states ln(A^B) = B ln(A):
ln((x + 3)^2) - ln(x^3)
4. Use the property of logarithms that allows us to rewrite the square of a term as (term)^2:
ln((x + 3)^2) - ln((x^3)^1)
5. Finally, apply the property of logarithms that states ln(A) - ln(B) = ln(A / B):
ln((x + 3)^2 / x^3)
Therefore, the condensed logarithmic expression can be written as ln((x + 3)^2 / x^3), with a coefficient of 1.