Rewrite the expression as a single logarithm.
1/4 In x+5[In (x-2)-3/10 In(x+2)]
a. In(5x(x-2)/6(x+2)
b. In (4sqrx (x-2)^5/sqr (x+2)^3
c. In(4sqrx(x-2)^5/ sqr(x+3)^5
the correct answer is b. In (4sqrx (x-2)^5/sqr (x+2)^3
show all steps
Apologies for the confusion earlier. Here are the correct steps to rewrite the expression as a single logarithm:
Given expression: 1/4 ln(x) + 5[ln(x-2) - (3/10)ln(x+2)]
Step 1: Distribute the scalar 5 to the terms inside the brackets:
= 1/4 ln(x) + 5ln(x-2) - (15/10)ln(x+2)
Step 2: Simplify the coefficients inside the logarithms:
= 1/4 ln(x) + ln(x-2)^5 - (3/2)ln(x+2)
Step 3: Combine the terms inside the logarithms using the exponent and quotient rules of logarithms:
= ln(x)^(1/4) + ln((x-2)^5) - ln((x+2)^(3/2))
Step 4: Simplify the exponents:
= ln(sqrt(x)) + ln((x-2)^5) - ln((x+2)^(3/2))
Step 5: Use the power rule of logarithms to combine the terms inside each logarithm:
= ln(sqrt(x)(x-2)^5) - ln((x+2)^(3/2))
Step 6: Use the quotient rule of logarithms to combine the terms:
= ln((sqrt(x)(x-2)^5)/((x+2)^(3/2)))
Step 7: Simplify the expression under the logarithm by removing the square root from the denominator:
= ln((sqrt(x)(x-2)^5)/(sqrt(x+2)^3))
Step 8: Simplify further by combining like terms inside the logarithm:
= ln((sqrt(x)(x-2)^5)/(sqrt(x+2))^3)
Step 9: Rationalize the denominator by multiplying both the numerator and denominator by (sqrt(x+2))^3:
= ln((sqrt(x)(x-2)^5*(sqrt(x+2))^3)/(x+2)^3)
Step 10: Simplify the expression inside the logarithm:
= ln((x(x-2)^5(x+2))/(x+2)^3)
Therefore, the expression can be rewritten as:
ln((4sqrt(x)(x-2)^5)/(sqrt(x+2))^3)
So the correct answer is b. ln(4sqrt(x)(x-2)^5/sqrt(x+2)^3)
To rewrite the expression as a single logarithm, we will use the properties of logarithms to combine terms.
Given expression: 1/4 ln(x) + 5[ln(x-2) - (3/10)ln(x+2)]
Step 1: Use the power rule of logarithms to move the coefficients outside the logarithm.
= ln(x)^(1/4) + ln(x-2)^5 - ln(x+2)^(3/10)
Step 2: Use the product rule of logarithms to combine the terms inside each logarithm.
= ln(x)^(1/4) + ln((x-2)^5) - ln((x+2)^(3/10))
Step 3: Use the quotient rule of logarithms to combine the terms.
= ln(x)^(1/4) + ln(((x-2)^5)/((x+2)^(3/10)))
Therefore, the expression as a single logarithm is:
ln(x)^(1/4) + ln(((x-2)^5)/((x+2)^(3/10)))