EXPONENTIAL FUNCTION If 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 = 2x Example: Evaluate the function y = 4xat the given values of x. a) x = 2 b) x = -3 c) x = 0
PROPERTIES OF EXPONENTIAL FUNCTION y = bx
The domain is the set of all real numbers.
The range is the set of positive real numbers.
The y – intercept of the graph is 1.
The x – axis is an asymptote of the graph.
The function is one – to – one.
The graph of the function y = bx y 1 x o
EXAMPLE 1: Graph the function y = 3x y 1 x o
EXAMPLE 2: Graph the function y = (1/3)x y 1 x o
NATURAL EXPONENTIAL FUNCTION: f(x) = ex y 1 x o
LOGARITHMIC FUNCTION For 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 = by Examples of logarithmic functions: a) y = log3 x b) f(x) = log6 x c) y = log2 x
EXAMPLE 1: Express in exponential form: EXAMPLE 2: Express in logarithmic form:
PROPERTIES OF LOGARITHMIC FUNCTIONS
The domain is the set of positive real numbers.
The range is the set of all real numbers.
The x – intercept of the graph is 1.
The y – axis is an asymptote of the graph.
The function is one – to – one.
The graph of the function y = logb x y x o 1
EXAMPLE 1: Graph the function y = log3 x y 1 x o
EXAMPLE 2: Graph the function y = log1/3 x y x 1 o
PROPERTIES OF EXPONENTS If a and b are positive real numbers, and m and n are rational numbers, then the following properties holds true:
To solve exponential equations, the following property can be used: bm = bn if and only if m = n and b > 0, b ≠ 1 EXAMPLE 1: Simplify the following: EXAMPLE 2: Solve for x:
PROPERTIES OF LOGARITHMS If M, N, and b (b ≠ 1) are positive real numbers, and r is any real number, then
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:
EXAMPLE 2: Express as a single logarithm:
NATURAL LOGARITHM Natural 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:
CHANGE-OF-BASE FORMULA EXAMPLE: Use common logarithms and natural logarithms to find each logarithm:
Solving Exponential Equations Guidelines: 1. Isolate the exponential expression on one side of the equation. 2. Take the logarithm of each side, then use the law of logarithm to bring down the exponent. 3. Solve for the variable. EXAMPLE: Solve for x:
Solving Logarithmic Equations Guidelines: 1. Isolate the logarithmic term on one side of the equation; you may first need to combine the logarithmic terms. 2. Write the equation in exponential form. 3. Solve for the variable. EXAMPLE 1: Solve the following:
EXAMPLE: Solve for x:
Application: (Exponential and Logarithmic Equations) The growth rate for a particular bacterial culture can be calculated using the formula B = 900(2)t/50, where B is the number of bacteria and t is the elapsed time in hours. How many bacteria will be present after 5 hours? How many hours will it take for there to be 18,000 bacteria present in the culture in example (1)? A fossil that originally contained 100 mg of carbon-14 now contains 75 mg of the isotope. Determine the approximate age of the fossil, to the nearest 100 years, if the half-life of carbon-14 is 5,570 years.
In a town of 15,000 people, the spread of a rumor that the local transit company would go on strike was such that t hours after the rumor started, f(t) persons heard the rumor, where experience over time has shown that a) How many people started the rumor? b) How many people heard the rumor after 5 hours? 5. A sum of $5,000 is invested at an interest rate of 5% per year. Find the time required for the money to double if the interest is compounded (a) semi-annually (b) continuously.