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.