Exponential And Logarithmic Functions

Geometric Series

How Can You Find The Sum Of The Terms Of A Finite Or Infinite Geometric Sequence? Convergent series: a series whose sum exists Divergent series: an infinite series whose sum does not exist Geometric series: the sum of the terms of a geometric sequence Definition of a Geometric Series Geometric sequence: an=a1·rn−1a_n=a_1\cdot r^{n-1} The sum of terms of a geometric sequence is called a geometric series. a1+a1r+a1r2+a1r3+...+a1rn−1a_1+a_1r+a_1r^2+a_1r^3+...+a_1r^{n-1} They can be finite or infinite. Sum of the Terms of a Finite Geometric Series, SnS_n Sn= a1(1−rn) (1r) , r≠1S_n=\frac{a_1(1-r^n)}{(1r)},\:r\ne1 What is the sum of the first six terms of the geometric sequence? 1, 4, 16, 64, ... n = 6 First term a1=1a_1=1 Common ratio r = 4 S6= a1(1−rn) (1−r) = 1(1−46) 1−4 = 1(1−4096) −3 =− 4095 −3 =−1365S_6=\frac{a_1(1-r^n)}{(1-r)}=\frac{1(1-4^6)}{1-4}=\frac{1(1-4096)}{-3}=-\frac{4095}{-3}=-1365 Summation Notation (Notes) Sum of an Infinite Geometric Series Sn= a1(1−rn) (1−r) , r≠1S_n=\frac{a_1(1-r^n)}{(1-r)},\:r\ne1 If ∣r∣<1\midr\mid<1 , the series converges to S. S= a1 (1−r) , r≠1S=\frac{a_1}{(1-r)},\:r\ne1 If ∣r∣>1\midr\mid>1,\:the\:ser , the series diverges.

Solving Exponential Equations by Rewriting the Base

How Can You Solve Equations That Have Variable Exponents? Exponential equation: an equation in which a variable occurs in the exponent How to Solve Exponential Equations with a Common Base An equation in which a variable occurs in the exponent is called an exponential equation. For b > 0, b ≠ 1, bx=byb^x=b^y if and only if x = y. How to Solve Exponential Equations by Rewriting Bases Rewrite each side of the equation so that they have the same base. Set exponents equal to each other. Solve the equation and check solutions. Substitution Property: A quantity may be substituted for its equal.

Solving Logarithmic Equations using Technology

How Can You Solve Logarithmic Equations Both Numerically And Graphically? Change of Base Formula log824= log24 log8 \log_824=\frac{\log24}{\log8} Change of base formula: If a, b, and x are positive real numbers, and neither a nor b is 1, then: logax= logx loga \log_ax=\frac{\log x}{\log a} How to Solve a Logarithmic Equation by Graphing To solve logarithmic equations by graphing: Write a system of equations. Rewrite each logarithm using the change of base formula. Graph the system. Identify the solution. How to Solve a Logarithmic Equation with a Variable on Both Sides Write a system of two equations. Graph the system. Identify the solution. How to Solve an Equation Containing Two Logarithms Write a system of equations. Rewrite each logarithm using the change of base formula. Graph the system. Identify the solution.

Properties of Logarithms

How Can You Use Properties Of Logarithms To Rewrite Or Evaluate Logarithmic Expressions? Power property of logarithms: the log of a number raised to a power equals the value of the power times the log of the number raised to a power of 1 Product property of logarithms: the log of a product equals the sum of the logs of each factor Quotient property of logarithms: the log of a quotient equals the difference of the log of the dividend and the log of the divisor Product Property of Logarithms Product property: logbxy=logbx+logby\log_bxy=\log_bx+\log_by Quotient Property of Logarithms Quotient property: logb x y =logbx−logby\log_b\frac{x}{y}=\log_bx-\log_by Power Property of Logarithms Power property: logbxr=rlogbx\log_bx^r=r\log_bx

Evaluating Logarithmic Expressions

How Do Inverse Relationships Help You Evaluate Logarithmic Expressions And Solve Equations? Exponentiate (an expression): to convert a logarithmic expression to an exponential expression How to Evaluate Logarithms To evaluate log x, think: "b to what power equals x?" Evaluate: log464\log_464 Evaluate: log4 1 16 \log_4(\frac{1}{16}) Special logarithms (b > 0, b ≠ 1): logb1=0\log_b1=0 logbb=1\log_bb=1

Graphing Exponential Functions

What Are The Key Features Of The Graph Of An Exponential Function? Exponential decay: a function in the form f(x) = bx, where x is an independent variable and b is a constant such that b > 0 and b ≠ 1 Exponential function: a decay in which the amount multiplies by the same factor between 0 and 1 for equal increases in time Exponential growth: a growth in which the amount multiplies by the same factor greater than 1 for equal increases in time Characteristics of Exponential Growth Functions: y=bx (b>1)y=b^x\:(b>1) When the base b > 1, the function y=bxy=b^x Has a domain of all real numbers. Has a range y > 0. Is always increasing. Has a y-intercept of 1. The Horizontal Asymptote of y=bx (b>1)y=b^x\:(b>1) The horizontal asymptote: The graph approaches the line y = o. The reason: As x approaches negative infinity, y approaches 0 because y=bxy=b^x gets smaller and smaller. Exponential Decay: y=bx (0<b<1)y=b^x\:(0 When the base b is between 0 and 1, the function y=bx:y=b^x: Has a domain of all real numbers. Has a range y > 0. Is always decreasing. Has a y-intercept of 1. y=bxy=b^x is a real-valued function only if b > 0. Translations of Exponential Functions Vertical translation: y=bx+ky=b^x+k shifts the graph of y=bxy=b^x up k units if k > 0, and down k units if k < 0. Horizontal translation: y=bx−hy=b^{x-h} shifts the graph of y=bxy=b^x right h units if h > 0, and left h units if h < 0.

Graphing Logarithmic Functions

What Methods Can You Use To Graph The Inverses Of Exponential Functions? Common logarithm: a logarithm with a base of 10; the notation log10 or log is used for a common logarithm Logarithm: the exponent of a given base that results in the power having a certain value; the notation logbm is used for a logarithm, where b is the base such that b > 0 and b ≠ 1 and m is the value of the base raised to the power Logarithmic function: a function that can be written in the form y = logbx, where b > 0 and b ≠ 1, which is the inverse of the exponential function y = bx Logarithms A logarithm is the exponent of a given base that results in the power having a certain value. If ba=xb^a=x , then logbx=a\log_bx=a , where b > 0 and b ≠ 1. A common logarithm is a logarithm with a base of 10. log10x=logx\log_{10}x=\log x Example: log28\log_28 The base b is 2. The value of the base raised to the power a is 8. So, log28=3\log_28=3 because 23=82^3=8 Logarithmic Functions A logarithmic function is a function that can be written in the form y=logbxy=\log_bx , where b > 0 and b ≠ 1. A logarithmic function y=logbxy=\log_bx is the inverse of the exponential function y=bxy=b^x How to Identify the Domain, Range, and Asymptote of a Logarithmic Function Domain of ƒ(x)=4xf(x)=4^x : All real numbers Range of ƒ−1(x)=log4x:f^{-1}(x)=\log_4x: All real numbers Range of ƒ(x)=4xf(x)=4^x : y > 0 Domain of ƒ−1(x)=log4x:f^{-1}(x)=\log_4x: x > 0 Horizontal Asymptote of ƒ(x)=4xf(x)=4^x : y = 0 Vertical Asymptote of ƒ−1(x)=log4xf^{-1}(x)=\log_4x : x = 0 Effect of the Base on the Graph of a Logarithmic Function For graphs of y=logbxy=\log_bx : All graphs pass through (1,0). b0=1, so logb1=0b^0=1,\:so\:\log_b1=0 When b > 1, the function increases as x increases. When 0 < b < 1, the function decreases as x increases When b > 1 and x > 1, the function with the smaller base increases faster. Transformations of the Parent Function: ƒ(x)=logb(x)f(x)=\log_b(x) How does the graph of g(x)=alogb(x−h)+kg(x)=alog_b(x-h)+k differ form the graph of its parent function, ƒ(x)=logb(x)f(x)=\log_b(x) ? ∣a∣>1\mida\mid>1 Stretch 0<∣a∣<10<\mida\mid<1 Compression a < 0: Reflection in the x-xis h > 0: Horizontal translation h units to the right h < 0: Horizontal translation h units to the left k > 0: Vertical translation k units up k < 0: vertical translation k units down

Base e

What Other Common Bases Are Used For Exponential And Logarithmic Functions? Natural logarithm: the power to which e must be raised to equal x Properties of the Graph of y=exy=e^x D: All real numbers R: y > 0 Horizontal Asymptote at y = o Natural Logarithms The natural logarithm of a number, x, is the power to which e must be raised to equal x. The natural logarithm of x,logex\log_ex , is denoted ln x. lnex=x\ln e^x=x lne=1 because e1=e\ln e=1\:because\:e^1=e ln1=0, because e0=1\ln1=0,\:because\:e^0=1 Properties of the Graph of y=lnxy=\ln x D: x > 0 R: All real numbers Vertical Asymptote at x = 0 Product and Quotient Properties of lnx\ln x Product property of natural logarithms: lnxy=lnx+lny\ln xy=\ln x+\ln y Quotient property of natural logarithms: ln x y =lnx−lny\ln\frac{x}{y}=\ln x-\ln y Power Property of lnx\ln x Power property of natural logarithms: lnxy=y lnx\ln x^y=y\:\ln x

Solving Equations using Properties of Logarithms

Can The Properties Of Logarithms Be Used To Solve Equations? Always check the solutions in the original equation. Logarithmic and Exponential Functions Logarithmic functions, ƒ(x)=logbxf(x)=\log_bx : Are one-to-one. Have inverse functions Real-World Applicatoin In chemistry, logarithms are key to calculating the pH of a solution.