[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571809-105":53,"doc-detail-571809-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","exponentials-and-logarithms-cheat-sheet","Exponentials and Logarithms - Cheat Sheet","","Exponentials and Logarithms Cheat Sheet summarizes key ideas behind exponential functions, including their graph behavior and differentiation rules. It highlights the special role of the constant e (approximately 2.71878) and provides derivative identities for general exponential forms. The sheet then connects exponential models to real contexts like population growth and radioactive decay, including a worked pesticide-decay example. It covers logarithms as inverses, laws of logarithms, solving equations using logarithms, natural logs, and interpreting logarithmic transformations of non-linear data.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/exponentials-and-logarithms-cheat-sheet/571809/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/exponentials-and-logarithms-cheat-sheet/571809.png","ImageObject",442,249,{"name":88,"@type":89},"jayni","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",9,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What defines an exponential function and how do its graphs behave?","Question",{"text":109,"@type":110},"Functions of the form a^(x)=c are exponential functions. The cheat sheet emphasizes that you should know their graph shapes and that the gradient-function graph mirrors the function’s shape for these forms.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Why is the constant e important when differentiating exponentials?",{"text":114,"@type":110},"When a is approximately 2.71878, the gradient function is exactly the same as the original function. The sheet notes that this exact constant is represented by the letter e.",{"name":116,"@type":107,"acceptedAnswer":117},"How are logarithms related to exponential functions and how are they used to solve equations?",{"text":118,"@type":110},"Logarithms are the inverses of exponential functions. The sheet states that log_a(x) = y is equivalent to a^y = x, and it shows solving equations by “taking logs” of both sides.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571809,1792002590,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":4},3985747859343,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Exponentials and Logarithms Cheat Sheet  \nExponential functions  \nFunctions of the form 􀀡(􀀣) = 􀀧!, where 􀀧 is a constant, are called exponential functions. You should become familiar with these functions and the shapes of their graphs.  \nFor instance, table below shows an example of values for 􀀩 = 2!.  \n\n| 􀀣 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 􀀩 | 1 8 | 1 4 | 1 2 | 1 | 2 | 4 | 8 |\n\nThe graph of 􀀩 = 2! is a smooth curve that looks like this:  \n􀀩  \n􀀣  \n􀀡 = 􀀤􀀡  \nExponential functions of the form 􀀡(􀀣) = 􀀧! have a special property. The graphs of their gradient functions are a similar shape to the graphs of the function themselves. When the value of a is approximately equal to 2.71878, the gradient function is exactly the same as the original function. The exact value of this is represented by the letter e.  \nFor all real values of 􀁋:  \n• If 􀁌(􀁋) = 􀁍􀀻 then 􀁌′(􀁋) = 􀁍􀀻  \n• If 􀁏 = 􀁍􀀻 then 􀀼􀀼􀀽􀀻 = 􀁍􀀻  \nA similar result holds for functions such as 􀀯 \"!, 􀀯 \\#!and 􀀯 ~~ ~~\"!!.  \nFor all real values of 􀁋 and for any constant 􀁐:  \n• If 􀁌(􀁋) = 􀁍􀀾􀀻 then 􀁌′(􀁋) = 􀁐􀁍􀀾􀀻  \n• If 􀁏 = 􀁍􀀾􀀻 then 􀀼􀀼􀀽􀀻 = 􀁐􀁍􀀾􀀻  \nExample 1: Differentiate with respect to 􀀣 .  \na. 􀀯 $! b. 􀀯 \\#\"!! c. 3􀀯 %!  \n$!  \na. 􀀩 = 􀀯 ~~ ~~  \n􀀱􀀱􀀩􀀣 = 4􀀯$!  \nb. 􀀩 = 􀀯 \\#\"!!  \n􀀱􀀩   1 \\#&!  \n= − 􀀯 %  \n􀀱􀀣 2  \nUse the rule for differentiating 􀀯 2! with 􀀹 = 4  \nc. 􀀩 = 3􀀯 %!􀀱􀀩  \n= 2 ×􀀱􀀣  \nExponential modelling  \n􀀯 ! can be used to model situations such as population growth, where the rate of increase is proportional to the size of the population at any given moment. Similarly, 􀀯 \\#! can be used to model radioactive decay, where the rate of decrease is proportional to the number of atoms remaining.  \nExample 2:  \nThe density of a pesticide in a given section of field, P mg/m2 , can be modelled by the equation  \n􀀵 = 160􀀯 \\#'.'')*  \nwhere t is the time in days since the pesticide was first applied.  \na. Use this model to estimate the density of pesticide after 15 days.  \nAfter 15 days, 􀀷 = 15.  \n􀀵 = 160􀀯\\#'.'')×&\"  \n􀀵 = 146.2 mg/m2  \nb. Interpret the meaning of the value 160 in this model.  \nWhen 􀀷 = 0, 􀀵 = 160􀀯, = 160, so 160 mg/m2 is the initial density of pesticide in the  \nfield.  \nc. Show that--.* = 􀀹􀀵, where 􀀹 is a constant, and state the value of 􀀹 .  \n􀀵 = 160􀀯 \\#'.'')*  \n-.  If 􀀩 = 􀀯 2! then-1 = 􀀹􀀯 2!  \n-* = −0.96􀀯\\#'.'')*, so 􀀹 = −0.96 -!  \nd. Interpret the significance of the sign of your answer to part c.  \nAs 􀀹 is negative, the density of the pesticide is decreasing (there is exponential decay)  \ne. Sketch the graph of P against t.  \n􀀵  \n160  \n0  \nLogarithms  \n􀀷  \nThe inverses of exponential functions are called logarithms.  \n• log/ 􀀾 = 􀀣 is equivalent to 􀀧! = 􀀾 (􀀧 ≠ 1)  \nExample 3: Write each statement as a logarithm.  \n!  \na. 3! = 9 b. 2\" = 128 c. 64\" = 8  \na. 3! = 9, so log 0 9 = 2  \nb. 2\" = 128, so log% 128 = 7  \nLaws of logarithms  \nExpressions involving more than one logarithm can be rearranged or simplified. The laws of logarithms:  \n• log/ 􀀣 + log/ 􀀩 = log/ 􀀣􀀩 (the multiplication law)  \n• log/ 􀀣 − log/ 􀀩 = log/ 1!~~ ~~ (the division law)  \n• log/ (􀀣 2 ) = 􀀹log/ 􀀣 (the power law)  \nYou should also recognise the following special cases:  \n• log/ &! = log/ (􀀣 \\#& ) = −log/ 􀀣 (the power law when 􀀹 = −1 )  \n• log/ 􀀧 = 1 (􀀧 > 0, 􀀧 ≠ 1)  \n• log/ 1 = 0 (􀀧 > 0, 􀀧 ≠ 1)  \nExample 4: Write as a single logarithm.  \na. log 0 6 + log 0 7  \n= log 0 (6 × 7)  \n= 42  \nb. log% 15 − log% 3  \n= log%(15 ÷ 3)  \n= log% 5  \nPure Year 1  \nc. 2log\" 3 + 3log\" 2  \n2log\" 3 = log\" (3% ) = log\" 9 3log\" 2 = log\" (20) = log\" 8 log\" 9 + log\" 8 = log\" 72  \nd. log&' 3 − 4log&' F12G  \n4log&' F12G = log&' F12G$ = log&' F116G  \nlog&' 3 − log&' F116G = log&' F3 ÷ 116G = log&' 48  \nSolving equations using logarithms  \nYou can use logarithms and your calculator to solve equations of the form 􀀧! = 􀁈 . You can also solve more complicated equations by ‘taking logs’ of both sides.  \n• Whenever 􀀡(􀀣) = 􀁉 (􀀣), log/ 􀀡 (􀀣) = log/ 􀁉 (􀀣)  \nExample 5: Solve the following equations, giving y","cbCaihfPglbK5bbG","https://ap.wps.com/l/cbCaihfPglbK5bbG","pdf",207102,"English","# Exponential functions\n## Derivatives and the constant e\n## Exponential modelling\n# Logarithms\n## Inverse relationship and logarithm equivalence\n## Laws of logarithms\n## Solving equations using logarithms\n## Working with natural logarithms\n## Logarithms and non-linear data","[{\"question\":\"What defines an exponential function and how do its graphs behave?\",\"answer\":\"Functions of the form a^(x)=c are exponential functions. The cheat sheet emphasizes that you should know their graph shapes and that the gradient-function graph mirrors the function’s shape for these forms.\"},{\"question\":\"Why is the constant e important when differentiating exponentials?\",\"answer\":\"When a is approximately 2.71878, the gradient function is exactly the same as the original function. The sheet notes that this exact constant is represented by the letter e.\"},{\"question\":\"How are logarithms related to exponential functions and how are they used to solve equations?\",\"answer\":\"Logarithms are the inverses of exponential functions. The sheet states that log_a(x) = y is equivalent to a^y = x, and it shows solving equations by “taking logs” of both sides.\"}]","Exponentials and Logarithms - Cheat Sheet | PDF",1791923794]