Calculus Volume 2 · free preview
Integrals, Exponential Functions, and Logarithms
The natural logarithm can be defined as the accumulation integral of 1 over t from 1 to x, deriving all its properties from the integral and recovering the exponential as its inverse, a rigorous foundation built on the fundamental theorem.
1. Defining the logarithm as an integral
Defining the logarithm as an integral
2. Properties from the integral
Properties from the integral
3. The exponential as inverse
The exponential as inverse
4. The number e and the hyperbola
The number e and the hyperbola
5. The defining integral
According to the integral definition, what is ln x for positive x?
6. Logarithm of one
From the integral definition, what is ln 1?
7. Product law proof move
The proof that ln(ab) = ln a + ln b from the integral definition splits the integral at which point?
8. Exponential from logarithm
In the rigorous construction, how is the natural exponential e^x defined?
9. Reciprocal law from the integral
The reciprocal law ln(1/b) = -ln b follows from the integral definition by which observation?
10. Derivative from FTC
By the second part of the fundamental theorem, the derivative of ln x equals the integrand of its defining integral. What function is that? Answer like 1/x.
11. Power law substitution
The power law ln(a^p) = p ln a is proved from the integral definition by which substitution?
12. Log of e
The number e is defined by ln e = 1. As a decimal approximation to two places, what is e? Answer like 2.72.
13. Integral of reciprocal from 1 to e
The integral from 1 to e of dt/t equals ln e by definition. What number is that? Answer with a number.
14. Behavior at zero
As x approaches 0 from the right, ln x = integral from 1 to x of dt/t tends to what value? Answer with the single word negative infinity.
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