Evaluate -28.9 ln(x) + 64.449 when x = 90.1.

Step-by-step solution

1. 1

   Substitute and note the order of operations. The expression becomes

   $-28.9\ln(90.1)+64.449$

   The logarithm is evaluated first, then multiplied by $-28.9$, and only then is $64.449$ added. In particular this is not $\ln$ of anything times $x$, and the $+64.449$ is outside the logarithm.

2. 2

   Compute the natural logarithm to enough digits. To six decimal places,

   $\ln(90.1)=4.500920$

   A useful cross-check: $\ln 90.1=\ln 9.01+\ln 10\approx 2.198335+2.302585=4.500920$. ✓ Because the multiplier $-28.9$ is large, every digit you drop here is magnified about 29 times, so do not round to four decimals yet.

3. 3

   Multiply by the coefficient.

   $-28.9\times 4.500920=-130.076588$

   The sign is negative because $-28.9<0$ and $\ln(90.1)>0$.

4. 4

   Add the constant term.

   $-130.076588+64.449=-65.627588\approx -65.63$

5. 5

   See what premature rounding would have cost. Using $\ln(90.1)\approx 4.5017$ instead gives $-28.9\times 4.5017=-130.099$ and a final value of $-65.65$. That is off by about $0.023$ — enough to move the answer to $-65.65$ instead of $-65.63$, and the reason to carry six digits through the multiplication and round only at the end.