if f(x) = 2x - x ln x
show that f(e^3) = -e^3
I know you have to substitute e^3 into the function, so Ive sone ths so far:
2e^3 - e^3 ln e^3
What is the next step?
2e^3 - e^3*lne^3,
lne^3 in exponential form:
e^a = e^3,
Since the 2 sides and bases are equal,
the exponents are also equal.
Therefore,
a = 3. lne^3 = 3.
Substitute e^3 3 for lne^3:
2e^3 - e^3 * 3,
Rearrange 2nd term:
2e^3 - 3e^3 = -e^3.
REMEMBER:
lne = 1,
lne^2 = 2,
lne^3 = 3,
lne^4 = 4.
CORRECTION:
Substitute 3 for lne^3.
The next step is to simplify the expression by evaluating the natural logarithm of e^3.
The natural logarithm function, ln(x), is the inverse of the exponential function e^x. Therefore, ln(e^3) is equal to 3.
Substituting this value back into the expression, we have:
2e^3 - e^3 ln(e^3)
= 2e^3 - e^3 * 3
Now, we can simplify further:
= 2e^3 - 3e^3
= -e^3
Hence, we have proved that f(e^3) = -e^3.