Evaluate, rounding to two decimal places, if necessary
a) ln 5
b) ln e²
c) (ln e)²
I am not going to simply do all your calculator practice for you.
is it
a) 1.61
b) 2
c) 1
To evaluate these expressions, we first need to understand the mathematical function ln, which represents the natural logarithm.
a) ln 5:
The natural logarithm is the inverse of the exponential function with the base e, where e is an irrational number approximately equal to 2.71828. Using a calculator or a mathematical software, you can find the value of ln 5. Alternatively, you can use this step-by-step guide to calculate an approximate value.
1. Start by calculating ln(1 + r) using the series expansion for small values of r. Since 5 is a small value compared to e, this method will give us a good approximation.
ln(1 + r) = r - (r² / 2) + (r³ / 3) - (r⁴ / 4) + ...
In this case, r = 4, so the formula becomes:
ln 5 = 4 - (4² / 2) + (4³ / 3) - (4⁴ / 4) + ...
ln 5 = 4 - (16 / 2) + (64 / 3) - (256 / 4) + ...
ln 5 ≈ 4 - 8 + 21.33 - 64 = -46.67
Therefore, ln 5 is approximately -46.67 when rounded to two decimal places.
b) ln e²:
Since the natural logarithm is the inverse of the exponential function, ln e is equal to 1. Thus, we can rewrite ln e² as ln (e * e), which simplifies to ln e² = 2 * ln e = 2 * 1 = 2.
Therefore, ln e² is equal to 2.
c) (ln e)²:
As mentioned earlier, ln e is equal to 1. Therefore, (ln e)² = 1² = 1.
Hence, (ln e)² is equal to 1.