{"id":4652,"date":"2012-10-30T05:58:54","date_gmt":"2012-10-30T05:58:54","guid":{"rendered":"http:\/\/SchoolTutoring.com\/help\/?p=4652"},"modified":"2014-12-02T08:31:58","modified_gmt":"2014-12-02T08:31:58","slug":"properties-of-logarithmic-functions","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/properties-of-logarithmic-functions\/","title":{"rendered":"Properties of Logarithmic Functions"},"content":{"rendered":"

<\/strong>A logarithmic function is basically the inverse of an exponential function.<\/p>\n

Take a >0 and a\u22601<\/em><\/strong>. \u00a0Then the inverse of the function f:R\u00e0R defined fy f(x)=ax<\/sup> is loga<\/sub>x <\/em><\/strong>which is called as logarithm of x<\/em><\/strong> with base a<\/em><\/strong>.<\/p>\n

So the definition of logarithm is as follows.<\/p>\n

loga<\/sub>x = y \u2194ay<\/sup> = x.<\/em><\/strong><\/p>\n

Example:<\/strong><\/p>\n

23<\/sup>=8 ==> log2<\/sub>8 = 3.<\/p>\n

Properties of logarithm:<\/strong><\/p>\n

(1)\u00a0\u00a0 <\/strong>loga<\/sub> (mn) = loga<\/sub> m + loga<\/sub> n<\/strong><\/p>\n

loga<\/sub> mn = x ==> mn = ax<\/sup><\/p>\n

loga<\/sub> m= y ==>m = ay<\/sup><\/p>\n

loga<\/sub> n = z ==> n = az<\/sup>
\nFrom all the equations,<\/p>\n

ax<\/sup>= ay <\/sup>. az<\/sup><\/p>\n

ax<\/sup>= ay+z<\/sup><\/p>\n

x=y+z<\/p>\n

loga<\/sub> (mn) = loga<\/sub> m + loga<\/sub> n<\/p>\n

(2)\u00a0\u00a0 <\/strong>loga<\/sub> (m\/n) = loga<\/sub> m – loga<\/sub> n<\/strong><\/p>\n

loga<\/sub> m\/n = x ==> m\/n = ax<\/sup><\/p>\n

loga<\/sub> m= y ==>m = ay<\/sup><\/p>\n

loga<\/sub> n = z ==> n = az<\/sup>
\nFrom all the equations,<\/p>\n

ax<\/sup>= ay <\/sup>\/ az<\/sup><\/p>\n

ax<\/sup>= ay-z<\/sup><\/p>\n

x=y-z<\/p>\n

loga<\/sub> (mn) = loga<\/sub> m -loga<\/sub> n<\/p>\n

(3)\u00a0\u00a0 <\/strong>loga<\/sub> mn<\/sup> = n loga<\/sub> m<\/strong><\/p>\n

loga<\/sub> mn<\/sup> = x ==>mn<\/sup> = ax<\/sup><\/p>\n

loga<\/sub> m= y ==>m = ay<\/sup><\/p>\n

From all the equations,<\/p>\n

ax<\/sup>= (ay <\/sup>)n<\/sup><\/p>\n

loga<\/sub> mn<\/sup> = n loga<\/sub> m<\/p>\n

(4)\u00a0\u00a0\u00a0 Change of base formula:<\/strong><\/p>\n

While using logarithms in solving equations, this formula is mostly used to change the bases.<\/p>\n

loga<\/sub> m = logb<\/sub> m\/logb<\/sub> a<\/strong><\/p>\n

Let loga<\/sub> m= x ==>ax<\/sup>=m<\/p>\n

Logb<\/sub> m= y ==>by<\/sup>=m<\/p>\n

logb<\/sub> a= z==> bz<\/sup>=a<\/p>\n

So we get,<\/p>\n

ax<\/sup>= by<\/sup><\/p>\n

(bz<\/sup>)x<\/sup>=by<\/p>\n

zx = y<\/p>\n

x = y\/z<\/p>\n

loga<\/sub> m = logb<\/sub> m\/logb<\/sub> a<\/p>\n

Do you also need help with Standardized Tests? Take a look at our Test Prep tutoring services.<\/a><\/p>\n

SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Quebec visit: Tutoring in Quebec. <\/a><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

A logarithmic function is basically the inverse of an exponential function. Take a >0 and a\u22601. \u00a0Then the inverse of the function f:R\u00e0R defined fy f(x)=ax is logax which is called as logarithm of x with base a. So the definition of logarithm is as follows. logax = y \u2194ay = x. Example: 23=8 ==> […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[178,704,707,1060,1412],"class_list":["post-4652","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-base","tag-function","tag-functionlogarithm","tag-logarithm","tag-properties"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/4652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=4652"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/4652\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=4652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=4652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=4652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}