Contract the expressions. That is, use the properties of logarithms to write each expression as a single logarithm with a coefficient of 1.
text((a) ) ln\(3\)-2ln\(4\)+ln\(8\)
text((b) )ln\(3\)-2ln\(4+8\)
text((c) )ln\(3\)-2(ln\(4\)+ln\(8\))
To contract the expressions and write each expression as a single logarithm with a coefficient of 1, we will use the properties of logarithms:
1. The product rule: log_a(x) + log_a(y) = log_a(xy)
2. The quotient rule: log_a(x) - log_a(y) = log_a(x/y)
3. The power rule: log_a(x^n) = n * log_a(x)
Let's work through each expression:
(a) ln(3) - 2ln(4) + ln(8)
First, we can apply the power rule to the second term:
2ln(4) = ln(4^2) = ln(16)
Now we can rewrite the expression:
ln(3) - ln(16) + ln(8)
Next, we can apply the quotient rule to combine the first two terms:
ln(3/16) + ln(8)
Finally, we can apply the product rule to combine the two terms:
ln(3/16 * 8) = ln(24/16) = ln(3/2)
So, the expression (a) can be contracted to ln(3/2).
(b) ln(3) - 2ln(4 + 8)
First, we can simplify the term inside the logarithm:
ln(4 + 8) = ln(12)
Now we can rewrite the expression:
ln(3) - 2ln(12)
Using the product rule, we can simplify further:
ln(3) - ln(12^2) = ln(3) - ln(144)
Finally, we combine the terms using the quotient rule:
ln(3/144) = ln(1/48)
So, the expression (b) can be contracted to ln(1/48).
(c) ln(3) - 2(ln(4) + ln(8))
First, we can apply the power rule inside the parentheses:
ln(4) + ln(8) = ln(4) + ln(8^2) = ln(4) + ln(64)
Now we can rewrite the expression:
ln(3) - 2(ln(4) + ln(64))
Using the product rule, we can simplify further:
ln(3) - ln(4^2 * 64^2) = ln(3) - ln(16 * 4096) = ln(3) - ln(65536)
Finally, we can combine the terms using the quotient rule:
ln(3/65536)
So, the expression (c) can be contracted to ln(3/65536).
Therefore, the contracted expressions are:
(a) ln(3/2)
(b) ln(1/48)
(c) ln(3/65536)