{"id":3578,"date":"2019-07-09T13:18:04","date_gmt":"2019-07-09T13:18:04","guid":{"rendered":"http:\/\/codethataint.com\/blog\/?p=3578"},"modified":"2019-07-09T15:46:01","modified_gmt":"2019-07-09T15:46:01","slug":"logarithm-basics","status":"publish","type":"post","link":"https:\/\/codethataint.com\/blog\/logarithm-basics\/","title":{"rendered":"Logarithm Basics"},"content":{"rendered":"

Logarithmic and Exponential Form<\/strong>
\nAny number can be written in Logarithmic and Exponential Form<\/p>\n

Logarithmic Form<\/em>
\nlogb<\/sub>x = a<\/p>\n

Exponential Form<\/em>
\nx = ba<\/sup><\/p>\n

Logarithmic Function and Exponential function are inverse function to each other<\/strong>
\nA function which is written in exponential form can be written into logarithmic form given both have the same base.<\/p>\n

y = log3<\/sub>x <\/p>\n

is equivalent to\t<\/p>\n

3y<\/sup> = x<\/p>\n

Let take a number 34<\/sup><\/strong><\/p>\n

34<\/sup> = 3 x 3 x 3 x 3 = 81<\/p>\n

log3<\/sub>81 = 4<\/p>\n

Here
\n3 is Base
\n4 is exponent<\/p>\n

What is the exponent of the below statement? What is the value of x?<\/em><\/p>\n

log5<\/sub>125 = x
\n5 x 5 x 5 = 125
\nx = 3<\/p>\n

log4<\/sub>1 = x
\nx = 0<\/p>\n

Why Know Logarithms?<\/strong>
\nLet us take the set of below numbers<\/p>\n

1
\n10
\n1000
\n100000(105<\/sup>)
\n100000000(108<\/sup>)
\n100000000000000(1014<\/sup> )<\/p>\n

When we try to plot the numbers we won’t be able to accommodate the small and the higher values since the variations are huge and the difference between the first and last number is huge. In such a case, we can represent the numbers in terms of logarithms with respect to some other decimal value. Now the above numbers could be written as below.<\/p>\n

log10<\/sub>1 = 0
\nlog10<\/sub>1 = 1
\nlog10<\/sub>1000 = 3
\nlog10<\/sub>x = 5
\nlog10<\/sub>x = 8
\nlog10<\/sub>x = 14<\/p>\n

Now it is easy to accommodate 0,1,3,5,8,14 in the graph.<\/p>\n

Practical application of Logarithms?<\/strong>
\nThe best practical usage of the logarithm is the Richter scale which is used to measure the earthquake. Richter scale is a logarithmic scale with base 10. Let’s say we there are earthquakes in 3 locations as below
\nIndia – 6.0
\nThailand – 7.0
\nJapan – 9.0<\/p>\n

Now the difference of intensity between the earthquake in India and Thailand is 10 Times Stronger
\nThe difference of intensity between the earthquake in India and Japan is 1000 Times Stronger<\/p>\n

Fundamental properties<\/strong><\/p>\n

1st Property<\/strong>
\nlogb<\/sub>x = a
\nx = ba<\/sup>
\nlogb<\/sub>x = ba<\/sup><\/p>\n

2nd Property<\/strong>
\nlogb<\/sub>b = x
\nlogbx<\/sup> = b
\nx = 1<\/p>\n

log44<\/sup> = 1
\nlog2525<\/sup> = 1<\/p>\n

3rd Property<\/strong>
\nlogb<\/sub>bn<\/sup> = x
\nbx<\/sup> = bn<\/sup>
\nx = n<\/p>\n

log11<\/sub>113<\/sup> = 3
\nlog4<\/sub>45<\/sup> = 5<\/p>\n

    \n
  1. logb<\/sub>x = ba<\/sup><\/li>\n
  2. logb<\/sub>b = 1<\/li>\n
  3. logb<\/sub>bn<\/sup> = n<\/li>\n<\/ol>\n

    Condition<\/strong><\/p>\n

  4. logb<\/sub>a – the value of a(argument) should be greater than 0 – Incorrect<\/li>\n
  5. logb<\/sub>0 is undefined and logb<\/sub>(-ve no) is incorrect – Incorrect<\/li>\n
  6. logb<\/sub>1= 0 is correct <\/li>\n
  7. logb<\/sub>(1\/2)= -1 is correct<\/li>\n
  8. log1<\/sub>a is incorrect- base cannot be 1<\/li>\n
  9. log(-ve)<\/sub>a is incorrect- base cannot be -ve number<\/li>\n

    Change of Base Rule<\/strong><\/p>\n

  10. logb<\/sub>a = logx<\/sub>a\/logx<\/sub>b where x>0, x != 1 and a,b >0 <\/li>\n

    log<\/strong> denotes log to the base 10 and ln<\/strong> or log e<\/sub><\/strong> represents natural log. <\/p>\n

    e – Maximum possible result after continuously compounding 100% growth over a time period<\/em><\/p>\n

    loga<\/sub>b x logb<\/sub>a = 1<\/p>\n

    Logarithm Rules<\/strong><\/p>\n

      \n
    1. logb<\/sub>(xy)= logb<\/sub>x + logb<\/sub>y<\/li>\n
    2. logb<\/sub>(x\/y)= logb<\/sub>x – logb<\/sub>y<\/li>\n
    3. logb<\/sub>an<\/sup> = nlogb<\/sub>a<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

      Logarithmic and Exponential Form Any number can be written in Logarithmic and Exponential Form Logarithmic Form logbx = a Exponential Form x = ba Logarithmic Function and Exponential function are inverse function to each other A function which is written in exponential form can be written into logarithmic form given both have the same base.… Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[288],"tags":[],"class_list":["post-3578","post","type-post","status-publish","format-standard","hentry","category-logarithms"],"_links":{"self":[{"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/posts\/3578","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/comments?post=3578"}],"version-history":[{"count":5,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/posts\/3578\/revisions"}],"predecessor-version":[{"id":3598,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/posts\/3578\/revisions\/3598"}],"wp:attachment":[{"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/media?parent=3578"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/categories?post=3578"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/codethataint.com\/blog\/wp-json\/wp\/v2\/tags?post=3578"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}