Find ln(1/(√(ab^3)) of lna = 2 and lnb = 3.
= ln1 - ln(ab)^(3/2)
= ln1 - lna^(3/2) + lnb^(3/2)
= 0 - 3/2(lna) + 3/2(lnb)
= -3/2(2) + 3/2(3)
= 1.5
--textbook answer is -5.5 ... where did I go wrong?
ln [ 1/{a^(1/2)b^(3/2)} ]
ln [a^-1/2 b^-3/2]
ln a^-1/2 + ln b^-3/2
-1/2 ln a -3/2 ln b
-1/2 (2) - 3/2 (3)
You made a mistake in the step where you applied the power rule for logarithms.
The correct step should be:
= ln1 - ln(ab^(3/2))
= ln1 - [lna + ln(b^(3/2))]
= ln1 - lna - (3/2)lnb
= 0 - 2 - (3/2)lnb
= -2 - (3/2)(3)
= -2 - (9/2)
= -2 - 4.5
= -6.5
So the correct answer is -6.5, not 1.5.
It seems like you made a mistake in the substitution step. Let's go through it again to find the error.
Given: ln(1/(√(ab^3))), lna = 2, and lnb = 3.
To find ln(1/(√(ab^3))), we can simplify it step by step.
Step 1: Simplify the expression inside the natural logarithm.
ln(1/(√(ab^3))) = ln(1/√a*b^3)
Step 2: Use the logarithmic property ln(x/y) = ln(x) - ln(y).
ln(1/√a*b^3) = ln(1) - ln(√a*b^3)
Step 3: Simplify ln(1).
ln(1/√a*b^3) = 0 - ln(√a*b^3)
Step 4: Apply the property ln(√x) = (1/2)ln(x).
ln(1/√a*b^3) = 0 - (1/2)ln(a*b^3)
Step 5: Substitute the given values for lna and lnb.
ln(1/√a*b^3) = 0 - (1/2)ln(a) - (1/2)*ln(b^3)
Since lna = 2 and lnb = 3, we substitute these values.
ln(1/√a*b^3) = 0 - (1/2)*2 - (1/2)*3
ln(1/√a*b^3) = 0 - 1 - (3/2)
ln(1/√a*b^3) = -1 - (3/2)
ln(1/√a*b^3) = -5/2
So, the correct answer is -5/2 or -2.5.
Please recheck your calculations to see where you went wrong in obtaining the solution of -5.5.