Question

$\mathrm{S}=30\left(1-\mathrm{e}^{(\ln 5 / 6) 5}\right)=30\left(1-\left(\frac{5}{6}\right)^{5}\right)$

Short answer

Since this is a multiplication and a natural logarithm, we can start by finding the logarithm of 5: $$ \ln 5 \approx 1.6094 $$ Now, divide this result by 6: $$ \frac{1.6094}{6} \approx 0.2682 $$ Next, multiply the result by 5: $$ 0.2682 \times 5 \approx 1.341 $$ #tag_title# Step 3: Evaluate the exponential term#tag_content# Now that we have the exponent, we can evaluate the e term: $$ e^{1.341} \approx 3.823 $$ This means that our expression is now essentially: $$ S = 30(1 - 3.823) $$ #tag_title# Step 4: Complete the subtraction and multiplication#tag_content# Finally, we can complete the subtraction inside the parentheses and then multiply with the factor 30: $$ 1 - 3.823 = -2.823 $$ Multiply this result by 30: $$ S = 30 \times (-2.823) \approx -84.69 $$ #tag_answer# The result of the expression is approximately -84.69.

Step-by-step solution

Identify the terms and operations

In this expression, we have several operations: exponentiation, subtraction, multiplication, and a natural logarithm. We'll deal with them one by one, following the order of operations. The given expression is: $$ S = 30(1 - e^{(\ln 5 / 6)5}) = 30(1 - (\frac{5}{6})^5) $$

Evaluate the exponent with the natural logarithm

First, we'll focus on the exponent inside of the e term: $$ (\ln 5 / 6) 5 $$