Topic 10 : Exponential and Logarithmic Functions Solving Exponential and Logarithmic Equations

Previously we studied…

Now we will study…

Exponential Function The function defined by is called an exponential function with base b and exponent x. The domain of f is the set of all real numbers.

Identify Exponential Functions Which of the following are exponential functions? y = 3 x y = x 3 y = 2(7) x y = 2(-7) x yes no

Identify the Base Identify the base in each of the following. y = 3 x y = 2(7) x y = 3a x y = 4 x - 3

Example The exponential function with base 2 is the function with domain (– ,  ). The values of f(x) for selected values of x follow:

More Examples The values of f(x) for selected values of x follow:

Recall Laws of Exponents Let a and b be positive numbers and let x and y be real numbers. Then,

Evaluate Exponential Functions y = 3 x for x = 4 y = 2(7) x for x = 3 y = -2(4 x ) for x = 3/2 y = 81 y = 686 y = -16

One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. For b>0 & b≠1 if b x = b y, then x=y Exponential Equations

Solve by equating exponents 4 3x = 8 x+1 (2 2 ) 3x = (2 3 ) x+1 rewrite w/ same base 2 6x = 2 3x+3 6x = 3x+3 x = 1 Check → 4 3*1 = = 64

Your turn! 2 4x = 32 x-1 2 4x = (2 5 ) x-1 4x = 5x-5 5 = x Be sure to check your answer!!!

But… what happens when the base are different? When you can’t rewrite using the same base, you can solve by taking a log of both sides

Exponential vs Logarithms We’ve discussed exponential equations of the form y = b x (b > 0, b ≠ 1) You may recall that y is called the logarithm of x to the base b, and is denoted log b x. –Logarithm of x to the base b y = log b x if and only if x = b y (x > 0)

Rewriting Logarithmic Equations:

Examples Solve log 3 x = 4 for x: Solution By definition, log 3 x = 4 implies x = 3 4 x = 81

Examples Solve log 16 4 = x for x: Solution log 16 4 = x is equivalent to 4 = 16 x 4 = (4 2 ) x or 4 1 = 4 2x from which we deduce that

Examples

Logarithmic Notation

Laws of Logarithms If m and n are positive numbers, then

Change base Formula

When you can’t rewrite using the same base, you can solve by taking a log of both sides 2 x = 7 log 2 x = log 7 x log 2 = log 7 x = ≈ 2.807

Practice…Practice…Practice