Calculus Volume 1 · free preview
Exponential and Logarithmic Functions
Exponential functions grow by constant multiplicative factors and are undone by logarithms, whose algebraic laws convert multiplication into addition and power rules, making them the working tools of growth models.
1. Exponential growth and decay
Exponential growth and decay
2. Logarithms as inverse functions
Logarithms as inverse functions
3. The laws of logarithms
The laws of logarithms
4. Solving exponential equations
Solving exponential equations
5. Why e is special
Which property distinguishes the natural exponential f(x) = e^x among all exponential functions?
6. Logarithm as inverse
Which equation says exactly the same thing as log base 2 of x equals 5?
7. The power law
Which expression equals ln(x^3) for positive x?
8. Solving an exponential equation
What is the exact solution of e^(2x) = 7?
9. Extraneous solutions
Solving log base 2 of x plus log base 2 of (x - 2) equals 3 produces the algebraic candidates x = 4 and x = -2. Which candidates are actual solutions?
10. Domain of a shifted logarithm
What is the domain of the function f(x) = ln(x + 2)?
11. Exponential value
The function f(x) = 2^x is evaluated at x = 5. What number results? Answer with a number.
12. Natural log of e
The natural logarithm undoes the exponential base e. What number is ln(e^3)? Answer with a number.
13. Solve a small exponential equation
The equation 5^x = 25 has an integer solution. What is it? Answer with a number.
14. Log of a product
Using ln 2 approximately 0.6931 and ln 3 approximately 1.0986, what is ln 6 rounded to four decimal places? Answer with a number.
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