[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-464503-105":59,"doc-detail-464503-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","2021-precalc-chapter-3-practice-test-qanda","2021 PreCalc Chapter 3 Practice Test QandA","","Worksheet-style practice and solution notes focused on exponentiation and logarithms, including conversions between exponential form and logarithmic form. The content repeatedly applies first-last-middle extraction, rewrites logarithmic expressions as exponential equations, and then solves for variables using algebraic steps. It also covers common log rules and transformations such as combining log terms, distributing and factoring, and using natural logarithms as equivalent forms. Several problems include domain checks, marking negative results as invalid for logarithms.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/2021-precalc-chapter-3-practice-test-qanda/464503/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/2021-precalc-chapter-3-practice-test-qanda/464503.png","ImageObject",300,407,{"name":92,"@type":93},"Mabel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-19","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does the document convert a logarithm equation into an exponential equation for solving?","Question",{"text":112,"@type":113},"It rewrites log_b(A)=C as b^C=A, then solves the resulting exponential equation using algebra.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the “first-last-middle” idea used in these examples?",{"text":117,"@type":113},"It identifies the base (first), the argument (middle), and the resulting value (last), then sets up the corresponding exponential or logarithmic form.",{"name":119,"@type":110,"acceptedAnswer":120},"Why are some derived solutions marked invalid in the document?",{"text":121,"@type":113},"Because negative numbers fall outside the domain of the logarithm function, so those solutions fail the original equation or simplified log form.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},464503,1791127737,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},7971474920318,"https://ap-avatar.wpscdn.com/avatar/74000ee537e6b0ca360?x-image-process=image/resize,m_fixed,w_180,h_180&k=1788430171632181674","A note on terminology: Zeros and roots are the same thing. If they are real, as opposed to complex, they are also 􀝔‐intercepts of your graph.  \nIn the log expression, log ୠ 64 ൌ 2, first is “b”, last is “2” and middle is “64.” We put these in an exponential expression, from left to right, to get: 􀢈 􀫛 ൌ 􀫟􀫝 .  \nIn the log expression, log ଺ 216 ൌ 􀝔 , first is “6”, last is “􀝔” and middle is “216.” We put these inan exponential expression, from left to right, to get: 􀫟􀢞 ൌ 􀫛􀫚􀫟 .  \nFirst‐last‐middle works this way too.  \nIn the exponential expression, 2ଷ ൌ 􀝔 , first is “2”, last is “􀝔” and middle is “3.” We put these in a logarithmic expression, from left to right, to get: 􀜔􀜗􀜏 􀫛 􀢞 ൌ 􀫜 .  \nIn the exponential expression, 2ିଶ ൌ ଵସ, first is “2”, last is “ଵସ” and middle is “െ2 .” We put these in a logarithmic expression, from left to right, to get: 􀜔􀜗􀜏􀫛 ቀ􀫚􀫝ቁ ൌ െ􀫛 .  \nIn the log expression, log ଵ଴ 10 ൌ 􀝔 , first is “10”, last is “􀝔” and middle is “10.” We put these inan exponential expression, from left to right, to get: 10௫ ൌ 10, then solve:  \nlog ଵ଴ 10 ൌ 􀝔 converts to: 10௫ ൌ 10  􀢞 ൌ 􀫚  \nIn the log expression, log ଷ√3 ൌ 􀝔 , first is “3”, last is “􀝔” and middle is “√3 .” We put these in an exponential expression, from left to right, to get: 3௫ ൌ √3, then solve:  \nlog ଷ√3 ൌ 􀝔 converts to: 3௫ ൌ √3   \nNote: √3 ൌ 3 ଵൗଶ  \n􀢞 ൌ  \n􀫚  \n􀫛  \nIn the log expression, log ଺ 1 ൌ 􀝔 , first is “6”, last is “􀝔” and middle is “1.” We put these in an exponential expression, from left to right, to get: 6௫ ൌ 1, then solve:  \nlog ଺ 1 ൌ 􀝔 converts to: 6௫ ൌ 1  􀢞 ൌ 􀫙  \nProblems 8 to 9: Exponentiation and taking logarithms are inverse operations, so when they both exist, with the same base, they cancel each other out.  \n􀫟 ሺ􀜔􀜗􀜏􀫟 􀫚􀫞ሻ ൌ 􀫚􀫞  \n􀜔􀜗􀜏 􀫠 ሺ􀫠 􀫚􀫡 ሻ ൌ 􀫚􀫡  \nNote: I like to use parentheses to make it easier to read a problem. For example, Problems 8 and 9 have values at multiple levels. By using parentheses, I can make it easier to see what is going on in the problem.  \nBack to “first‐last‐middle.”  \nIn the log expression, log ଵ଴ 1000 ൌ 􀝔 , first is “10”, last is “􀝔” and middle is “1000.” We put these in an exponential expression, from left to right, to get: 10௫ ൌ 1000, then solve:  \nlog ଵ଴ 1000 ൌ 􀝔 converts to: 10௫ ൌ 1000  􀢞 ൌ 􀫜  \nIn the log expression, log ଵ଴ ሺ10଻ሻ ൌ 􀝔 , first is “10”, last is “􀝔” and middle is “10଻ .” We put these in an exponential expression, from left to right, to get: 10௫ ൌ 10଻, then solve:  \nlog ଵ଴ 10଻ ൌ 􀝔 converts to: 10௫ ൌ 10଻  􀢞 ൌ 􀫠  \nNote that ln 􀝔 is equivalent to log ௘ 􀝔 . Then,  \nIn the log expression, log ௘ 􀝁 ൌ 􀝔 , first is “e”, last is “􀝔” and middle is “􀝁 .” We put these inan exponential expression, from left to right, to get: 􀝁 ௫ ൌ 􀝁 , then solve:  \nlog ௘ 􀝁 ൌ 􀝔 converts to: 􀝁 ௫ ൌ 􀝁  􀢞 ൌ 􀫚  \nlog ଶ 8􀝔 ൌ log ଶ 8൅log ଶ 􀝔 ൌ 􀫜൅􀜔􀜗􀜏 􀫛 􀢞  \nlog ହ ൬12􀝔5൰ ൌ log ହ 125 െ log ହ 􀝔 ൌ 􀫜െ􀜔􀜗􀜏 􀫞 􀢞  \nlog ௕ ሺ􀝕􀝖ସ ሻ ൌ log ௕ ሺ􀝕ሻ ൅log௕ ሺ􀝖ସ ሻ ൌ 􀜔􀜗􀜏 􀢈 􀢟൅􀫝 􀜔􀜗􀜏 􀢈 􀢠  \nlog ସ ൬􀝔~~ ~~~~ ~~6൰ ൌ log ସ ሺ􀝔 െ 6ሻ െlogସ ሺ􀝔ହ ሻ ൌ 􀜔􀜗􀜏 􀫝 ሺ􀢞െ􀫟ሻ െ􀫞 􀜔􀜗􀜏 􀫝 􀢞  \nlog ସ ሺ􀝔 െ 8ሻ െ log ସ ሺ􀝔 െ 4ሻ ൌ 􀜔􀜗􀜏 􀫝 ൬􀢞􀢞~~ ~~െ~~ ~~􀫡􀫝൰  \n3 log଺ 􀝔 ൅ 5 log଺ ሺ􀝔 െ 6ሻ ൌ log ଺ 􀝔 ଷ ൅log ଺ ሺ􀝔 െ 6ሻହ ൌ 􀜔􀜗􀜏 􀫟 ൣ􀢞􀫜 ሺ􀢞െ􀫟ሻ􀫞 ൧  \n4 log௫ 2 െlog௫ 8 ൌ log௫ 2ସ െlog௫ 2ଷ ൌ log௫ 22ସଷ ൌ 􀜔􀜗􀜏 􀢞 􀫛 Alternatively,  \n4 log௫ 2 െlog௫ 8 ൌ 4 log௫ 2 െlog௫ 2ଷ ൌ 4 log௫ 2 െ 3 log௫ 2 ൌ 􀜔􀜗􀜏 􀢞 􀫛  \nStarting equation: 4 ሺଵାଶ௫ሻ ൌ 64  \nTake the \"logସ \" of both sides: 1൅2x ൌ 3  \nSubtract 1: 2􀝔 ൌ 2  \nDivide by 2: 􀢞 ൌ 􀫚  \nStarting equation: 􀝁 ሺ௫ା଼ሻ ൌ ଵ௘ర  \nConvert exponent on the right: 􀝁 ሺ௫ା଼ሻ ൌ 􀝁 ିସ  \nTake the \"ln\" of both sides: x൅8 ൌ െ4  \nSubtract 8: 􀢞 ൌ െ􀫚􀫛  \nStarting equation:  \nTake the \"ln\" of both sides: Simplify:  \nDivide by ln 5:  \nSubtract 7:  \n5 ሺ௫ା଻ሻ ൌ 3  \nln 5 ሺ௫ା଻ሻ ൌ ln 3 ሺx൅7ሻln 5 ൌ ln 3  \nx൅7 ൌ  \n୪୬ ଷ  \n୪୬ ହ  \n􀜔􀜖 􀫜  \n􀢞 ൌ  െ􀫠􀜔􀜖 􀫞  \nNote: I tend to use ln in solving problems like this. However, you can use logs with any base. Other useful bases for this problem might be log, log3 or log5.  \nStarting equation:  \nTake the ln of both sides: Simplify:  \nSubtract 4:  \nStarting equation:  \nTake 3 to the power of both sides: Simplify:  \nAd","cbCaijPWw8Cl366D","https://ap.wps.com/l/cbCaijPWw8Cl366D","pdf",627654,15,"English","# Exponentiation and Logarithms Conversion\n## First-Last-Middle Notation\n## Exponential vs Logarithmic Form\n## Natural Log Equivalence (ln)\n# Solving Logarithm Equations\n## Inverse Operations and Cancellation\n## Algebraic Solving Steps\n## Combining, Distributing, and Factoring\n# Domain Checks for Log Functions\n## Invalid Negative Inputs","[{\"question\":\"How does the document convert a logarithm equation into an exponential equation for solving?\",\"answer\":\"It rewrites log_b(A)=C as b^C=A, then solves the resulting exponential equation using algebra.\"},{\"question\":\"What is the “first-last-middle” idea used in these examples?\",\"answer\":\"It identifies the base (first), the argument (middle), and the resulting value (last), then sets up the corresponding exponential or logarithmic form.\"},{\"question\":\"Why are some derived solutions marked invalid in the document?\",\"answer\":\"Because negative numbers fall outside the domain of the logarithm function, so those solutions fail the original equation or simplified log form.\"}]","2021 PreCalc Chapter 3 Practice Test QandA | PDF",1790768277,38]