This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are: - Exponential functions take the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis. - The natural exponential function is f(x) = ex. - Logarithmic functions are the inverses of exponential functions and take the form f(x) = logbx. Their graphs stretch upwards and have asymptotes at the y-axis. - Properties of exponents and logarithms include the change of base formula. - Exponential and logarithmic equations can be solved by isolating the exponential/log

1. EXPONENTIAL AND LOGARITHMIC FUNCTIONS

2. EXPONENTIAL FUNCTIONIf x and b are real numbers such that b > 0 and b ≠ 1, then f(x) = bx is an exponential function with base b.Examples of exponential functions: a) y = 3x b) f(x) = 6x c) y = 2xExample: Evaluate the function y = 4xat the given values of x. a) x = 2 b) x = -3 c) x = 0

3. PROPERTIES OF EXPONENTIAL FUNCTION y = bx The domain is the set of all real numbers.

4. The range is the set of positive real numbers.

5. The y – intercept of the graph is 1.

6. The x – axis is an asymptote of the graph.

7. The function is one – to – one. The graph of the function y = bxy1xo

8. EXAMPLE 1: Graph the function y = 3xy1xo

9. EXAMPLE 2: Graph the function y = (1/3)xy1xo

10. NATURAL EXPONENTIAL FUNCTION: f(x) = exy1xo

11. LOGARITHMIC FUNCTIONFor all positive real numbers x and b, b ≠ 1, the inverse of the exponential function y = bx is the logarithmic functiony = logbx.In symbol, y = logb x if and only if x = byExamples of logarithmic functions: a) y = log3 x b) f(x) = log6 x c) y = log2 x

12. EXAMPLE 1: Express in exponential form:EXAMPLE 2: Express in logarithmic form:

13. PROPERTIES OF LOGARITHMIC FUNCTIONS The domain is the set of positive real numbers.

14. The range is the set of all real numbers.

15. The x – intercept of the graph is 1.

16. The y – axis is an asymptote of the graph.

17. The function is one – to – one. The graph of the function y = logb xyxo1

18. EXAMPLE 1: Graph the function y = log3 xy1xo

19. EXAMPLE 2: Graph the function y = log1/3 xyx1o

20. PROPERTIES OF EXPONENTSIf a and b are positive real numbers, and m and n are rational numbers, then the following properties holds true:

21. To solve exponential equations, the following property can be used:bm = bn if and only if m = n and b > 0, b ≠ 1EXAMPLE 1: Simplify the following:EXAMPLE 2: Solve for x:

22. PROPERTIES OF LOGARITHMSIf M, N, and b (b ≠ 1) are positive real numbers, and r is any real number, then

23. Since logarithmic function is continuous and one-to-one, every positive real number has a unique logarithm to the base b. Therefore,logbN = logbM if and only if N = M EXAMPLE 1: Express the ff. in expanded form:

24. EXAMPLE2: Express as a single logarithm:

25. NATURAL LOGARITHMNatural logarithms are to the base e, while common logarithms are to the base 10. The symbol ln x is used for natural logarithms.EXAMPLE: Solve for x: