Unit 5 Exponential, Logarithmic, Piecewise Functions

Power Rule

(a^m)^n=a^mn

Exponential Growth and Decay

-Exponential growth is modeled by functions of the form f(x)=ab^x, where a>0 and b>1. -Exponential decay is modeled by functions of the form f(x)=ab^x, where a>0 and 0<b<1.

Properties of Logarithms

-logb 1=0 -logb b=1 -logb b^x=x b^(logb x)=x

Solving Exponential Equations Using a Common Base

1. Rewrite the equation so that both sides have the same base: b^M=b^N. 2. Set the exponents equal to one another: M=N. 3. Solve for the variable.

Solving Logarithmic Equations Using the Definition of Logarithms

1. Use the rules of logarithms to rewrite the equation in the form logb M=c 2. Convert the equation to exponential form: logb M=c-->b^c=M 3. Solve for the variable 4. Check the solution to ensure only values for which M>0 are included.

Solving Logarithmic Equations Using the One-to-One Property

1. Write the equation in the form logb M=logb N. 2. Set the arguments equal to one another: M=N. 3. Solve for the variable. 4. Check the solution to ensure only values for which M>0 and N>0 are included.

Other Formulas for Logarithms

1/a^n=a^-n ^nsqrta^m=a^m/n a^y(a)=1/y loga (x)=b is equal to x=a^b

Change on Base Formula Example

4^p=9 log4 4^p=log4 9 p=log4 9 p=In 9/In 4 p=1.584962501...

Exponential Function Examples

9^x=27^2 Both sides do not have a common base, but they can be rewritten with a common base. The number 3 is the smallest natural number that is a common base. 9=3^2 27=3^3 (3^2)^x=(3^3)^2 Simplify. remember, to simplify a base with an exponent raised to an exponent, multiply the exponents. 3^2x=3^6 2x=6 x=3 Solve 100^x=0.001 100^x=0.001 (10^2)^x=(1/1000) (10)^2x=(10)^-3 2x=-3 x=-3/2 The solution is x=-3/2. Solve 27^x=81.3^x 27^x=81.3^x (3^3)^x=3^4*3^x 3^3x=3^4+x 3x=4+x 2x=4 x=2

Logarithmic Function

A logarithmic function is the inverse of the exponential function f(x)=b^x, where b>0 and b does not equal 1. The logarithmic function is written f(x)=logb x (b is small). The base of a logarithmic function is the same as the base of the corresponding exponential function, and is restricted to positive real numbers not equal to 1. For a base b, where b>0 and b does not equal 1, y=logb x if and only if x=b^y. For f(x)=logb x, the domain is (0,infinity) and the range is (-infinity,infinity). Since the logarithmic function is the inverse of the exponential function, the domain of the logarithmic function is the range of the exponential function and the range of the logarithmic function is the domain of the exponential function.

Natural Logarithm

A natural logarithm is the inverse of the natural exponential function f(x)=e^x. The natural logarithm is written f(x)=loge x or, more commonly, f(x)=In x.

The Natural Exponential Function and E

Although any number b, where b>0 and b does not equal 1, can be used as the base of the function f(x)=b^x, one of the most useful choices for b is the number e. Like the number pi, the number e is irrational and hence nonrepeating. There are several methods used to approximate the value of the irrational number, e. One method expresses e as an infinite sum of fraction with factorials in the denominator.

Exponential Function

An exponential function has the form f(x)=b^x, where b is a positive real number not equal to 1. If b>0, the function increases as x increases and exhibits what is called exponential growth.

Vertical and Horizontal Reflections of Exponential Functions

Consider the parent exponential function f(x)=b^x, where b is a positive real number not equal to 1. g(x)=-b^x : Reflection of the graph of the function f(x) over the x-axis g(x)=b^-x : Reflection of the graph of the function f(x) oover the y-axis

Vertical and Horizontal Translations of Exponential Functions

Consider the parent exponential function f(x)=b^x, where b is a positive real number not equal to 1. g(x)=b^x+h, for h>0 : Translation of the graph of the function f(x)h units to the left g(x)=b^x+h, for h>0 : Translation of the graph of the function f(x)h units to the right g(x)=b^x+h, for k>0 : Translation of the graph of the function f(x)k units up g(x)=b^x+h, for k>0 : Translation of the graph of the function f(x)k units down.

Common Logarithm Example

Evaluate log 1/100. log 1/100=p log10 1/100=p 10^p=1/100 10^p=10^-2 p=-2 Therefore, log 1/100=-2.

Change of Base Example

Evaluate log1/16 8 log1/16 8=p (1/16)^p=8 (2^-4)^p=2^3 2^-4p=2^3 -4p=3 p=-3/4

Properties of Exponential and Logarithmic Functions, b>0, b does not equal 1

Exponential Function: Function Form-f(x)=b^x Domain-(-infinity,infinity) Range-(0,infinity) Asymptote-x-axis X-Intercept-None Y-Intercept-(0,1) Logarithmic Function: Function Form-f(x)=logb x Domain-(0,infinity) Range-(-infinity,infinity) Asymptote-y-axis X-Intercept-(1,0) Y-Intercept-None

Quadratic Form of an Exponential Equation Example

Exponential equations such as e^2x-5e^x-14=0 have two terms, e^2x and e^x, with variable exponents. Since the two exponents, 2x and x, are not the same, the form of the equation needs to be changed. This exponential equation can be rewritten in quadratic form using a substitution process. Let w=e^x and rewrite the exponential equation. e^2x-5e^x-14=0 (e^x)^2-5e^x-14=0 w^2-5w-14=0 Solve the quadratic equation by factoring, and then apply the Zero Product Property. w^2-5w-14=0 (w-7)(w+2)=0 w=7 or w=-2 Use the fact that w=e^x to write equations that will enable you to find x. 7=e^x or -2=e^x Solve each of these equations by taking the natural log of each side of the equation. In 7=In(e^x) In 7=x or In (-2)=In(e^x) In (-2)=x The answer x=In(-2) is not a solution since it is undefined. The only solutiion is x=In 7. In order for an exponential equation to be quadratic in nature, one of the exponents must be twice the other.

Decimal Example

Find an approximation to 6 decimal places of log6 100. log6 100=log100/log6=2/log6=2.570194

Evaluating Logarithm Example

For example, to evaluate the expression log3 81, first use the expression to write a logarithmic equation. log3 81=p Then write the logarithmic equation in exponential form. 3^p=81 Write the right-hand side of the resulting equation as a power, using the same base as that in the expression on the left. 3^p=3^4 Since the bases are equal, the resulting exponents must also be equal. p=4 Therefore, log3 81=4.

Solving Exponential Equations Using Logarithms

If both sides of the equation cannot be written with a common base, use the following steps: 1. Isolate the exponential expression 2. Take the log of each side of the equation 3. Simplify using the rules for logarithms 4. Solve for the variable.

Properties of Natural Logarithms

In 1=0 In e=1 In e^x=x e^(In x)=x

Logarithmic Form vs. Exponential Form

Logarithmic Form: y=logb(x) where x>0 and b>0 Exponential Form: b^y=x where b does not equal 1

Condensing Logarithmic Expressions Example

Rewrite 2log2 x+3log2(x+4)-3log2(x-1) as a single logarithm. Apply the power rule first and then the product and quotient rules to write a single logarithm. 2log2 x+3log2(x+4)-3log2(x-1)= log2 x^2+log2(x+4)^3-log2(x-1)^3= log2(x^2(x+4)^3/(x-1)^3)

Different Bases Example

Solve 2^x+3=5^x-2 to six decimal places. 2^x+3=5^x-2 Take the common log of both sides. log 2^x+3=log 5^x-2 Apply the power rule for logarithms. (x+3)log 2=(x-2)log 5 Distributive Property xlog2+3log 2=xlog 5-2log 5 Subtraction Property 3log 2+2log 5=xlog 5-xlog 2 Factor out x. 3log 2+2log 5=x(log 5-log 2) Division Property 3log 2+2log 5/log 5-log 2=x Simplify using a calculator. 5.782354=x

Composite Logarithmic Functions

Solve log3(log4 x)=2 To solve this problem, you will need to convert to exponential form two times. Work from the outer function to the inner function log3(log4 x)=2 3^2=log4 x 9=log4 x 4^9=x x=262,144 Check.

Product Rule of Logarithms

States that the logarithm of a product is equal to the sum of the logarithms of each factor in the argument.

The Quotient Rule of Logarithms

States that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator in the argument.

Power Rule of Logarithms

States that the logarithm of an argument raised to a power is equal to the product of the power and the logarithm of the argument.

Logarithm Rules

Summarizing, to multiply powers with the same base, you add their exponents. To divide powers with the same base, you subtract their exponents. To raise a power to a power, you multiply their exponents. Product Rule - logb(AB)=logb A+logb B Quotient Rule - logb(A/B)=logb A-logb B Power Rule - logb A^r=rlogb A Corollary to the Power Rule: logb(1/A)=-logb A

Common Logarithm

The base of a logarithm can be any positive real number other than 1, but there are two bases that are used most often: 10 and the irrational number e. A common logarithm is a logarithm with a base of 10. When you see a logarithm without a base written, it is understood to be a common logarithm. In other words, log x=log10 x.

Natural Exponential Function

The function f(x)=e^x is known as the natural exponential function.

General Form of an Exponential Function

The general form of an exponential function is f(x)=ab^c(x-h)+k, where a does not equal 0, b is a positive real number not equal to 1, and c does not equal 0. -If b>1, the function is an increasing function. -If 0<b<1, the function is a decreasing function. -If a>0, the domain is (-infinity, infinity) and the range is (0, infinity). -If a<0, the domain is (-infinity, infinity) and the range is (-infinity, 0). In order to determine if a function is an exponential function, first look to see if the independent variable is in the exponent. The base of the function, b, must be a constant, positive real number not equal to 1.

Comparing Logarithmic Functions

Two logarithmic functions, f(x)=loga x and g(x)=logb x, where a>0 and b>0, may be compared in the following manner. If both bases are greater than 1, or both bases are between 0 and 1 and a>b, the following applies: -The graph of f lies above the graph of g over the interval (0,1). -The graphs intersect at x=1. -The graph of f lies below the graph of g when x>1. The base a is greater than 1 but base b is less than 1, the following applies: -The graph of f lies below the graph of g over the interval (0,1). -The graphs intersect at x=1. -The graph of f lies above the graph of g when x>1.

Product Rule

a^m*a^n=a^m+n

Quotient Rule

a^m/a^n=a^m-n

Expansion of e

e=1/0!+1/1!+1/2!+1/3!+1/4!+ e=s.7182818284...

Change of Base Formula

logb A=log A/log b= In A/In b

Exponential Growth and Decay Formula

y(t)=a*e^kt y=value at the time, t a=the value at the start k=rate of growth or decay t=time