Contract the expressions. That is, use the properties of logarithms to write each expression as a single logarithm with a coefficient of 1.
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((a) ) ln\(3\)-2ln\(4\)+ln\(8\)
((b) ln\(3\)-2ln\(4+8\)
(c) )ln\(3\)-2(ln\(4\)+ln\(8\))
To contract the expressions and write them as single logarithms with a coefficient of 1, we can use the properties of logarithms.
(a) ln(3) - 2ln(4) + ln(8):
First, let's start with the property of logarithms that states ln(a) - ln(b) = ln(a/b). Applying this property to the expression, we have:
ln(3) - 2ln(4) + ln(8) = ln(3) - ln(4^2) + ln(8)
Next, we can use the property ln(a) + ln(b) = ln(ab) to combine ln(4^2) and ln(8):
ln(3) - ln(4^2) + ln(8) = ln(3) + ln(8/4^2)
Simplifying further, we have:
ln(3) + ln(8/4^2) = ln(3 * 8/4^2)
Simplifying the numerator, denominator, and exponent, we get:
ln(3 * 8/4^2) = ln(24/16)
Finally, we can simplify the fraction:
ln(24/16) = ln(3/2)
Therefore, the contracted expression is ln(3/2).
(b) ln(3) - 2ln(4+8):
Using the same property as above, we can rewrite the expression as:
ln(3) - ln((4+8)^2)
Simplifying the exponent, we get:
ln(3) - ln(12^2)
Further simplifying, we have:
ln(3) - ln(144)
Therefore, the contracted expression is ln(3/144).
(c) ln(3) - 2(ln(4) + ln(8)):
Using the property ln(a) + ln(b) = ln(ab), we can distribute the coefficient of 2 to both ln(4) and ln(8):
ln(3) - 2(ln(4) + ln(8)) = ln(3) - 2ln(4) - 2ln(8)
Using the properties from part (a), we can simplify the expression:
ln(3) - 2ln(4) - 2ln(8) = ln(3) - ln(4^2) - ln(8^2)
Further simplifying the exponents:
ln(3) - ln(4^2) - ln(8^2) = ln(3) - ln(16) - ln(64)
Therefore, the contracted expression is ln(3/16/64).