Characteristics of the Parent Function [latex]f(x) = b^x[\/latex]:<\/strong><\/p>\r\n For [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\r\n\r\n Exponential Growth ([latex]b > 1[\/latex]):<\/strong><\/p>\r\n\r\n Exponential Decay ([latex]0 < b < 1[\/latex]):<\/strong><\/p>\r\n\r\n Transformations of Exponential Graphs:<\/strong><\/p>\r\n General form: [latex]f(x) = ab^{x+c} + d[\/latex]<\/p>\r\n\r\n General Form:<\/strong> [latex]f(x) = ab^x[\/latex] where:<\/p>\r\n\r\n The Number [latex]e[\/latex]:<\/strong><\/p>\r\n The mathematical constant [latex]e \\approx 2.718282[\/latex] is defined as: [latex]e = \\lim_{n \\to \\infty}\\left(1 + \\frac{1}{n}\\right)^n[\/latex]<\/p>\r\n The natural exponential function [latex]f(x) = e^x[\/latex] has the same characteristics as other exponential functions with base [latex]b > 1[\/latex].<\/p>\r\n Finding Exponential Equations from Two Points:<\/strong><\/p>\r\n Case 1:<\/strong> If one point is [latex](0, a)[\/latex]:<\/p>\r\n\r\n Case 2:<\/strong> If neither point is [latex](0, a)[\/latex]:<\/p>\r\n\r\n Finding Equations from Graphs:<\/strong><\/p>\r\n\r\n For [latex]x > 0[\/latex], [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\r\n [latex]y = \\log_b(x)[\/latex] means [latex]b^y = x[\/latex]<\/p>\r\n The logarithm [latex]y[\/latex] is the exponent to which [latex]b[\/latex] must be raised to get [latex]x[\/latex].<\/p>\r\n Converting Between Forms:<\/strong><\/p>\r\n\r\n Common Logarithm:<\/strong> Base 10<\/p>\r\n\r\n Natural Logarithm:<\/strong> Base [latex]e[\/latex]<\/p>\r\n\r\n Evaluating Logarithms:<\/strong><\/p>\r\n To evaluate [latex]\\log_b(x)[\/latex], ask: \"To what exponent must [latex]b[\/latex] be raised to get [latex]x[\/latex]?\"<\/p>\r\n Examples:<\/p>\r\n\r\n Since logarithmic and exponential functions are inverses:<\/p>\r\n\r\n Characteristics of the Parent Function [latex]f(x) = \\log_b(x)[\/latex]:<\/strong><\/p>\r\n For [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\r\n\r\n Finding the Domain:<\/strong><\/p>\r\n The argument of a logarithm must be positive. For [latex]f(x) = \\log_b(\\text{expression})[\/latex]:<\/p>\r\n\r\n Transformations of Logarithmic Graphs:<\/strong><\/p>\r\n General form: [latex]f(x) = a\\log_b(x + c) + d[\/latex]<\/p>\r\n\r\n base<\/strong><\/p>\r\n The constant [latex]b[\/latex] in an exponential function [latex]f(x) = ab^x[\/latex] or in a logarithmic function [latex]f(x) = \\log_b(x)[\/latex]; must be positive and not equal to 1.<\/p>\r\n common logarithm<\/strong><\/p>\r\n A logarithm with base 10, written as [latex]\\log(x)[\/latex] instead of [latex]\\log_{10}(x)[\/latex]; used to measure phenomena like earthquakes (Richter scale), star brightness, and pH levels.<\/p>\r\n exponential decay<\/strong><\/p>\r\n A decrease based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]0 < b < 1[\/latex] in [latex]f(x) = ab^x[\/latex].<\/p>\r\n exponential function<\/strong><\/p>\r\n A function of the form [latex]f(x) = ab^x[\/latex] where [latex]a[\/latex] is any nonzero number, [latex]b > 0[\/latex], and [latex]b \\ne 1[\/latex].<\/p>\r\n exponential growth<\/strong><\/p>\r\n An increase based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]b > 1[\/latex] in [latex]f(x) = ab^x[\/latex].<\/p>\r\n growth factor<\/strong><\/p>\r\n The value [latex]b[\/latex] in an exponential growth function [latex]f(x) = ab^x[\/latex] where [latex]b > 1[\/latex]; also called the base or growth multiplier.<\/p>\r\n horizontal asymptote<\/strong><\/p>\r\n A horizontal line that the graph of a function approaches as the input approaches [latex]\\infty[\/latex] or [latex]-\\infty[\/latex]; for exponential functions [latex]f(x) = ab^{x+c} + d[\/latex], the horizontal asymptote is [latex]y = d[\/latex].<\/p>\r\n initial value<\/strong><\/p>\r\n The value [latex]a[\/latex] in an exponential function [latex]f(x) = ab^x[\/latex]; represents the output when [latex]x = 0[\/latex], so [latex]f(0) = a[\/latex].<\/p>\r\n logarithm<\/strong><\/p>\r\n The exponent to which a base must be raised to produce a given number; if [latex]b^y = x[\/latex], then [latex]\\log_b(x) = y[\/latex].<\/p>\r\n logarithmic function<\/strong><\/p>\r\n The inverse of an exponential function; a function of the form [latex]f(x) = \\log_b(x)[\/latex] where [latex]b > 0[\/latex] and [latex]b \\ne 1[\/latex].<\/p>\r\n natural exponential function<\/strong><\/p>\r\n The exponential function with base [latex]e[\/latex]; written as [latex]f(x) = e^x[\/latex] where [latex]e \\approx 2.718282[\/latex].<\/p>\r\n natural logarithm<\/strong><\/p>\r\n A logarithm with base [latex]e[\/latex], written as [latex]\\ln(x)[\/latex] instead of [latex]\\log_e(x)[\/latex]; commonly used in calculus and scientific applications.<\/p>\r\n one-to-one function<\/strong><\/p>\r\n A function in which each output value corresponds to exactly one input value; both exponential and logarithmic functions are one-to-one.<\/p>\r\n vertical asymptote<\/strong><\/p>\r\n A vertical line that the graph of a function approaches but never touches or crosses; for logarithmic functions [latex]f(x) = \\log_b(x + c)[\/latex], the vertical asymptote is [latex]x = -c[\/latex].<\/p>","rendered":" Characteristics of the Parent Function [latex]f(x) = b^x[\/latex]:<\/strong><\/p>\n For [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\n Exponential Growth ([latex]b > 1[\/latex]):<\/strong><\/p>\n Exponential Decay ([latex]0 < b < 1[\/latex]):<\/strong><\/p>\n Transformations of Exponential Graphs:<\/strong><\/p>\n General form: [latex]f(x) = ab^{x+c} + d[\/latex]<\/p>\n General Form:<\/strong> [latex]f(x) = ab^x[\/latex] where:<\/p>\n The Number [latex]e[\/latex]:<\/strong><\/p>\n The mathematical constant [latex]e \\approx 2.718282[\/latex] is defined as: [latex]e = \\lim_{n \\to \\infty}\\left(1 + \\frac{1}{n}\\right)^n[\/latex]<\/p>\n The natural exponential function [latex]f(x) = e^x[\/latex] has the same characteristics as other exponential functions with base [latex]b > 1[\/latex].<\/p>\n Finding Exponential Equations from Two Points:<\/strong><\/p>\n Case 1:<\/strong> If one point is [latex](0, a)[\/latex]:<\/p>\n Case 2:<\/strong> If neither point is [latex](0, a)[\/latex]:<\/p>\n Finding Equations from Graphs:<\/strong><\/p>\n For [latex]x > 0[\/latex], [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\n [latex]y = \\log_b(x)[\/latex] means [latex]b^y = x[\/latex]<\/p>\n The logarithm [latex]y[\/latex] is the exponent to which [latex]b[\/latex] must be raised to get [latex]x[\/latex].<\/p>\n Converting Between Forms:<\/strong><\/p>\n Cannot take the logarithm of a negative number or zero<\/p>\n Common Logarithm:<\/strong> Base 10<\/p>\n Natural Logarithm:<\/strong> Base [latex]e[\/latex]<\/p>\n Evaluating Logarithms:<\/strong><\/p>\n To evaluate [latex]\\log_b(x)[\/latex], ask: “To what exponent must [latex]b[\/latex] be raised to get [latex]x[\/latex]?”<\/p>\n Examples:<\/p>\n Since logarithmic and exponential functions are inverses:<\/p>\n Characteristics of the Parent Function [latex]f(x) = \\log_b(x)[\/latex]:<\/strong><\/p>\n For [latex]b > 0[\/latex], [latex]b \\ne 1[\/latex]:<\/p>\n Finding the Domain:<\/strong><\/p>\n The argument of a logarithm must be positive. For [latex]f(x) = \\log_b(\\text{expression})[\/latex]:<\/p>\n Transformations of Logarithmic Graphs:<\/strong><\/p>\n General form: [latex]f(x) = a\\log_b(x + c) + d[\/latex]<\/p>\n base<\/strong><\/p>\n The constant [latex]b[\/latex] in an exponential function [latex]f(x) = ab^x[\/latex] or in a logarithmic function [latex]f(x) = \\log_b(x)[\/latex]; must be positive and not equal to 1.<\/p>\n common logarithm<\/strong><\/p>\n A logarithm with base 10, written as [latex]\\log(x)[\/latex] instead of [latex]\\log_{10}(x)[\/latex]; used to measure phenomena like earthquakes (Richter scale), star brightness, and pH levels.<\/p>\n exponential decay<\/strong><\/p>\n A decrease based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]0 < b < 1[\/latex] in [latex]f(x) = ab^x[\/latex].<\/p>\n exponential function<\/strong><\/p>\n A function of the form [latex]f(x) = ab^x[\/latex] where [latex]a[\/latex] is any nonzero number, [latex]b > 0[\/latex], and [latex]b \\ne 1[\/latex].<\/p>\n exponential growth<\/strong><\/p>\n An increase based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]b > 1[\/latex] in [latex]f(x) = ab^x[\/latex].<\/p>\n growth factor<\/strong><\/p>\n The value [latex]b[\/latex] in an exponential growth function [latex]f(x) = ab^x[\/latex] where [latex]b > 1[\/latex]; also called the base or growth multiplier.<\/p>\n horizontal asymptote<\/strong><\/p>\n A horizontal line that the graph of a function approaches as the input approaches [latex]\\infty[\/latex] or [latex]-\\infty[\/latex]; for exponential functions [latex]f(x) = ab^{x+c} + d[\/latex], the horizontal asymptote is [latex]y = d[\/latex].<\/p>\n initial value<\/strong><\/p>\n The value [latex]a[\/latex] in an exponential function [latex]f(x) = ab^x[\/latex]; represents the output when [latex]x = 0[\/latex], so [latex]f(0) = a[\/latex].<\/p>\n logarithm<\/strong><\/p>\n The exponent to which a base must be raised to produce a given number; if [latex]b^y = x[\/latex], then [latex]\\log_b(x) = y[\/latex].<\/p>\n logarithmic function<\/strong><\/p>\n The inverse of an exponential function; a function of the form [latex]f(x) = \\log_b(x)[\/latex] where [latex]b > 0[\/latex] and [latex]b \\ne 1[\/latex].<\/p>\n natural exponential function<\/strong><\/p>\n The exponential function with base [latex]e[\/latex]; written as [latex]f(x) = e^x[\/latex] where [latex]e \\approx 2.718282[\/latex].<\/p>\n natural logarithm<\/strong><\/p>\n A logarithm with base [latex]e[\/latex], written as [latex]\\ln(x)[\/latex] instead of [latex]\\log_e(x)[\/latex]; commonly used in calculus and scientific applications.<\/p>\n one-to-one function<\/strong><\/p>\n A function in which each output value corresponds to exactly one input value; both exponential and logarithmic functions are one-to-one.<\/p>\n vertical asymptote<\/strong><\/p>\n A vertical line that the graph of a function approaches but never touches or crosses; for logarithmic functions [latex]f(x) = \\log_b(x + c)[\/latex], the vertical asymptote is [latex]x = -c[\/latex].<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/688"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/688\/revisions"}],"predecessor-version":[{"id":5208,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/688\/revisions\/5208"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/688\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=688"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=688"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=688"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=688"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\r\n \t
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\r\n \r\nTransformation<\/th>\r\n Form<\/th>\r\n Effect on Graph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n \r\n Vertical shift<\/td>\r\n [latex]f(x) = b^x + d[\/latex]<\/td>\r\n Shifts up [latex]d[\/latex] units if [latex]d > 0[\/latex], down if [latex]d < 0[\/latex]; asymptote becomes [latex]y = d[\/latex]; range becomes [latex](d, \\infty)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Horizontal shift<\/td>\r\n [latex]f(x) = b^{x+c}[\/latex]<\/td>\r\n Shifts left [latex]c[\/latex] units if [latex]c > 0[\/latex], right if [latex]c < 0[\/latex]; y-intercept becomes [latex](0, b^c)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical stretch<\/td>\r\n [latex]f(x) = ab^x[\/latex] where [latex]|a| > 1[\/latex]<\/td>\r\n Stretches vertically by factor of [latex]|a|[\/latex]; y-intercept becomes [latex](0, a)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical compression<\/td>\r\n [latex]f(x) = ab^x[\/latex] where [latex]0 < |a| < 1[\/latex]<\/td>\r\n Compresses vertically by factor of [latex]|a|[\/latex]; y-intercept becomes [latex](0, a)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Reflection over x-axis<\/td>\r\n [latex]f(x) = -b^x[\/latex]<\/td>\r\n Reflects across x-axis; range becomes [latex](-\\infty, 0)[\/latex]; y-intercept becomes [latex](0, -1)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Reflection over y-axis<\/td>\r\n [latex]f(x) = b^{-x}[\/latex]<\/td>\r\n Reflects across y-axis; equivalent to [latex]f(x) = \\left(\\frac{1}{b}\\right)^x[\/latex]; reverses growth\/decay<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n Exponential Functions<\/h3>\r\n
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Logarithmic Functions<\/h3>\r\n
\r\n\r\n
\r\n \r\nLogarithmic Form<\/th>\r\n Exponential Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n \r\n [latex]\\log_b(x) = y[\/latex]<\/td>\r\n [latex]b^y = x[\/latex]<\/td>\r\n<\/tr>\r\n \r\n [latex]\\log_2(8) = 3[\/latex]<\/td>\r\n [latex]2^3 = 8[\/latex]<\/td>\r\n<\/tr>\r\n \r\n [latex]\\log_5(25) = 2[\/latex]<\/td>\r\n [latex]5^2 = 25[\/latex]<\/td>\r\n<\/tr>\r\n \r\n [latex]\\log_{10}\\left(\\frac{1}{10000}\\right) = -4[\/latex]<\/td>\r\n [latex]10^{-4} = \\frac{1}{10000}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nCannot take the logarithm of a negative number or zero\r\n \r\n \t
\r\n \t
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Graphs of Logarithmic Functions<\/h3>\r\n
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\r\n\r\n
\r\n \r\nTransformation<\/th>\r\n Form<\/th>\r\n Effect on Graph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n \r\n Horizontal shift<\/td>\r\n [latex]f(x) = \\log_b(x + c)[\/latex]<\/td>\r\n Shifts left [latex]c[\/latex] units if [latex]c > 0[\/latex], right if [latex]c < 0[\/latex]; asymptote becomes [latex]x = -c[\/latex]; domain becomes [latex](-c, \\infty)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical shift<\/td>\r\n [latex]f(x) = \\log_b(x) + d[\/latex]<\/td>\r\n Shifts up [latex]d[\/latex] units if [latex]d > 0[\/latex], down if [latex]d < 0[\/latex]; asymptote remains [latex]x = 0[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical stretch<\/td>\r\n [latex]f(x) = a\\log_b(x)[\/latex] where [latex]|a| > 1[\/latex]<\/td>\r\n Stretches vertically by factor of [latex]|a|[\/latex]; x-intercept remains [latex](1, 0)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Vertical compression<\/td>\r\n [latex]f(x) = a\\log_b(x)[\/latex] where [latex]0 < |a| < 1[\/latex]<\/td>\r\n Compresses vertically by factor of [latex]|a|[\/latex]; x-intercept remains [latex](1, 0)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Reflection over x-axis<\/td>\r\n [latex]f(x) = -\\log_b(x)[\/latex]<\/td>\r\n Reflects across x-axis; domain [latex](0, \\infty)[\/latex] and asymptote [latex]x = 0[\/latex] unchanged<\/td>\r\n<\/tr>\r\n \r\n Reflection over y-axis<\/td>\r\n [latex]f(x) = \\log_b(-x)[\/latex]<\/td>\r\n Reflects across y-axis; domain becomes [latex](-\\infty, 0)[\/latex]; asymptote remains [latex]x = 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n Key Equations<\/h2>\r\n
\r\n\r\n
\r\n \r\nDescription<\/th>\r\n Equation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n \r\n General exponential function<\/td>\r\n [latex]f(x) = ab^x[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Transformed exponential function<\/td>\r\n [latex]f(x) = ab^{x+c} + d[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Natural exponential function<\/td>\r\n [latex]f(x) = e^x[\/latex] where [latex]e \\approx 2.718282[\/latex]<\/td>\r\n<\/tr>\r\n \r\n General logarithmic function<\/td>\r\n [latex]f(x) = \\log_b(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Transformed logarithmic function<\/td>\r\n [latex]f(x) = a\\log_b(x + c) + d[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Common logarithm<\/td>\r\n [latex]\\log(x) = \\log_{10}(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Natural logarithm<\/td>\r\n [latex]\\ln(x) = \\log_e(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Logarithmic-exponential equivalence<\/td>\r\n [latex]y = \\log_b(x) \\Leftrightarrow b^y = x[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Inverse property (exponential to log)<\/td>\r\n [latex]\\log_b(b^x) = x[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Inverse property (log to exponential)<\/td>\r\n [latex]b^{\\log_b(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Graphs of Exponential Functions<\/h3>\n
\n
\n
\n
\n\n
\n \nTransformation<\/th>\n Form<\/th>\n Effect on Graph<\/th>\n<\/tr>\n<\/thead>\n \n Vertical shift<\/td>\n [latex]f(x) = b^x + d[\/latex]<\/td>\n Shifts up [latex]d[\/latex] units if [latex]d > 0[\/latex], down if [latex]d < 0[\/latex]; asymptote becomes [latex]y = d[\/latex]; range becomes [latex](d, \\infty)[\/latex]<\/td>\n<\/tr>\n \n Horizontal shift<\/td>\n [latex]f(x) = b^{x+c}[\/latex]<\/td>\n Shifts left [latex]c[\/latex] units if [latex]c > 0[\/latex], right if [latex]c < 0[\/latex]; y-intercept becomes [latex](0, b^c)[\/latex]<\/td>\n<\/tr>\n \n Vertical stretch<\/td>\n [latex]f(x) = ab^x[\/latex] where [latex]|a| > 1[\/latex]<\/td>\n Stretches vertically by factor of [latex]|a|[\/latex]; y-intercept becomes [latex](0, a)[\/latex]<\/td>\n<\/tr>\n \n Vertical compression<\/td>\n [latex]f(x) = ab^x[\/latex] where [latex]0 < |a| < 1[\/latex]<\/td>\n Compresses vertically by factor of [latex]|a|[\/latex]; y-intercept becomes [latex](0, a)[\/latex]<\/td>\n<\/tr>\n \n Reflection over x-axis<\/td>\n [latex]f(x) = -b^x[\/latex]<\/td>\n Reflects across x-axis; range becomes [latex](-\\infty, 0)[\/latex]; y-intercept becomes [latex](0, -1)[\/latex]<\/td>\n<\/tr>\n \n Reflection over y-axis<\/td>\n [latex]f(x) = b^{-x}[\/latex]<\/td>\n Reflects across y-axis; equivalent to [latex]f(x) = \\left(\\frac{1}{b}\\right)^x[\/latex]; reverses growth\/decay<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n Exponential Functions<\/h3>\n
\n
\n
\n
\n
Logarithmic Functions<\/h3>\n
\n\n
\n \nLogarithmic Form<\/th>\n Exponential Form<\/th>\n<\/tr>\n<\/thead>\n \n [latex]\\log_b(x) = y[\/latex]<\/td>\n [latex]b^y = x[\/latex]<\/td>\n<\/tr>\n \n [latex]\\log_2(8) = 3[\/latex]<\/td>\n [latex]2^3 = 8[\/latex]<\/td>\n<\/tr>\n \n [latex]\\log_5(25) = 2[\/latex]<\/td>\n [latex]5^2 = 25[\/latex]<\/td>\n<\/tr>\n \n [latex]\\log_{10}\\left(\\frac{1}{10000}\\right) = -4[\/latex]<\/td>\n [latex]10^{-4} = \\frac{1}{10000}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n
\n
\n
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Graphs of Logarithmic Functions<\/h3>\n
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\n \nTransformation<\/th>\n Form<\/th>\n Effect on Graph<\/th>\n<\/tr>\n<\/thead>\n \n Horizontal shift<\/td>\n [latex]f(x) = \\log_b(x + c)[\/latex]<\/td>\n Shifts left [latex]c[\/latex] units if [latex]c > 0[\/latex], right if [latex]c < 0[\/latex]; asymptote becomes [latex]x = -c[\/latex]; domain becomes [latex](-c, \\infty)[\/latex]<\/td>\n<\/tr>\n \n Vertical shift<\/td>\n [latex]f(x) = \\log_b(x) + d[\/latex]<\/td>\n Shifts up [latex]d[\/latex] units if [latex]d > 0[\/latex], down if [latex]d < 0[\/latex]; asymptote remains [latex]x = 0[\/latex]<\/td>\n<\/tr>\n \n Vertical stretch<\/td>\n [latex]f(x) = a\\log_b(x)[\/latex] where [latex]|a| > 1[\/latex]<\/td>\n Stretches vertically by factor of [latex]|a|[\/latex]; x-intercept remains [latex](1, 0)[\/latex]<\/td>\n<\/tr>\n \n Vertical compression<\/td>\n [latex]f(x) = a\\log_b(x)[\/latex] where [latex]0 < |a| < 1[\/latex]<\/td>\n Compresses vertically by factor of [latex]|a|[\/latex]; x-intercept remains [latex](1, 0)[\/latex]<\/td>\n<\/tr>\n \n Reflection over x-axis<\/td>\n [latex]f(x) = -\\log_b(x)[\/latex]<\/td>\n Reflects across x-axis; domain [latex](0, \\infty)[\/latex] and asymptote [latex]x = 0[\/latex] unchanged<\/td>\n<\/tr>\n \n Reflection over y-axis<\/td>\n [latex]f(x) = \\log_b(-x)[\/latex]<\/td>\n Reflects across y-axis; domain becomes [latex](-\\infty, 0)[\/latex]; asymptote remains [latex]x = 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n Key Equations<\/h2>\n
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\n \nDescription<\/th>\n Equation<\/th>\n<\/tr>\n<\/thead>\n \n General exponential function<\/td>\n [latex]f(x) = ab^x[\/latex]<\/td>\n<\/tr>\n \n Transformed exponential function<\/td>\n [latex]f(x) = ab^{x+c} + d[\/latex]<\/td>\n<\/tr>\n \n Natural exponential function<\/td>\n [latex]f(x) = e^x[\/latex] where [latex]e \\approx 2.718282[\/latex]<\/td>\n<\/tr>\n \n General logarithmic function<\/td>\n [latex]f(x) = \\log_b(x)[\/latex]<\/td>\n<\/tr>\n \n Transformed logarithmic function<\/td>\n [latex]f(x) = a\\log_b(x + c) + d[\/latex]<\/td>\n<\/tr>\n \n Common logarithm<\/td>\n [latex]\\log(x) = \\log_{10}(x)[\/latex]<\/td>\n<\/tr>\n \n Natural logarithm<\/td>\n [latex]\\ln(x) = \\log_e(x)[\/latex]<\/td>\n<\/tr>\n \n Logarithmic-exponential equivalence<\/td>\n [latex]y = \\log_b(x) \\Leftrightarrow b^y = x[\/latex]<\/td>\n<\/tr>\n \n Inverse property (exponential to log)<\/td>\n [latex]\\log_b(b^x) = x[\/latex]<\/td>\n<\/tr>\n \n Inverse property (log to exponential)<\/td>\n [latex]b^{\\log_b(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n Glossary<\/h2>\n