Could you please explain why lne^2 = 2, e^lne^2 = e^2, e^lin2 = 2, and lne^e = e ? Thank you.
by definition,
ln(x) is the power of e you need to get x.
Just as log_10(100)=2 because 100=10^2
b^(log_b(N)) = log_b(b^N) = N
ln(e^e) = e*ln(e) = e*1 = 1
Certainly! Let's break down each expression one by one.
1. ln(e^2) = 2:
- To understand this, we need to know that ln denotes the natural logarithm and e is the base of this logarithm.
- The expression e^2 represents raising the constant e to the power of 2, which equals e * e = e^2.
- The natural logarithm ln is the inverse function of e^x. So, when we take ln(e^2), it cancels out the exponentiation, leaving us with the original value inside the parentheses, which is 2.
2. e^(ln(e^2)) = e^2:
- Here, we have ln(e^2) inside the parentheses.
- As we discussed earlier, ln is the inverse function of e^x, so applying e^x to an ln expression will cancel each other out.
- Hence, e^(ln(e^2)) simplifies to just e^2, which means raising e to the power of 2.
3. e^(ln(2)) = 2:
- In this case, ln(2) represents the natural logarithm of 2.
- Similar to the previous explanation, applying e^x to ln(2) will cancel each other out.
- Therefore, e^(ln(2)) simplifies to just 2, which means raising e to the power of ln(2) results in 2.
4. ln(e^e) = e:
- Here, e^e represents raising the constant e to the power of e.
- The natural logarithm ln is the inverse function of e^x. So, when we take ln(e^e), it cancels out the exponentiation, leaving us with the original value inside the parentheses, which is e.
In summary, these equations demonstrate the properties of the natural logarithm function and exponential function when applied to each other. The natural logarithm "undoes" the exponential function, and vice versa.