Exponential Functions • A function of the form y=abx, • where a=0, b>0 and b=1. • Characteristics 1. continuous and one-to-one 2. domain is the set of all real numbers 3. Range is either all real positive numbers or all real negative numbers depending on whether a is , or > 0 4. x-axis is a horizontal asymptote 5.y-intercept is at a 6. y=abx and y=a(1/b)x are reflections across the y-axis
Example 1 • Sketch the graph of y=2x. State the domain and range.
Example 2 • Sketch y=( )x. State the domain and range.
Exponential Growth & Decay • Exponential Growth: • Exponential function with base greater than one. • y=2(3x) • Exponential Decay: • Exponential function with base between 0 and 1 • y=4(1/3)x
Example 3-6 • Determine if each function is exponential growth or decay y=(1/5)x y=7(1.2)x y=2(5)x y=10(4/3)x
Steps to write an exponential function • 1. Use the y-intercept to find a • 2. Chose a second point on the graph to substitute into the equation for x and y. Solve for b. • 3. Write your equation in terms of y=abx (plug in a and b)
Example 5 • Write an exponential function using the points (0, 3) and (-1, 6)
Example 6 • Write an exponential function using the points (0, -18) and (-2, -2)
Example 11 • In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in 2004. • A. Write an exponential function of the form y=abx that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since 2000. • B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.
Exponential Equations • Exponential equation: • An equation in which the variables are exponents • Property of Equality If the base is a number other than 1 and the base is the same , then the two exponents equal each other. • 2x = 28 then x=8
Steps to Solve Exponential Equations/inequalities • 1. Rewrite the equation so all terms have like bases (you may need to use negative exponents) • 2. Set the exponents equal to each other • 3. Solve • 4. Plug x back in to the original equation to make sure the answer works
Example 3 • Solve 35x = 92x-1
Example 4 • Solve 42x = 8x-1
Example 7 • Solve
Example 11 • Solve
Example 13 • Solve
Logarithms with base b • Say: “Log of x base b is y”
Characteristics of Logarithmic Functions • 1. Inverse of the exponential function y=bx • 2.Continous and one-to-one • 3. Domain is all positive real numbers and range is ARN • 4. y-axis is an asymptote • 5. Contains (1,0), so x-intercept is 1
Helpful Hint • Since exponential and logarithmic functions are inverses if the bases are the same they “undo” each other…
Logarithmic Equations • Property of Equality • If b is a positive number other than 1, then if and only if x = y.
Example 9 • Solve
Example 10 • Solve
Example 11 • Solve
Logarithmic to Exponential Inequality If b > 1, x > 0 and logbx > y then x > by If b > 1, x > 0 and logbx < y then 0< x < by
Example 12 • Solve
Example 13 • Solve
Property of Inequality for Logarithmic Functions • If b>1, then if and only if x>y and if and only if x<y
Product Property • The logarithm of a product is the sum of the logarithm of its factors
Quotient Property • The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.
Power Property • The logarithm of a power is the product of the logarithm and the exponent
Common Logarithms • Logarithms with base 10 are common logs • You do not need to write the 10 it is understood • Button on calculator for common logs LOG
Examples: Use calculator to evaluate each log to four decimal places • 1. log 3 2. log 0.2 • 3. log 5 4. log 0.5
Solve Logarithmic Equations • Example 5: • The amount of energy E, in ergs, that an earthquake releases is related to is Richter scale magnitude M by the equation logE = 11.8 + 1.5M. The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much energy was released?
Example 6: • Find the energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to the tsunami.