{"id":186808,"date":"2025-01-26T09:03:45","date_gmt":"2025-01-26T09:03:45","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186808"},"modified":"2025-01-26T09:03:47","modified_gmt":"2025-01-26T09:03:47","slug":"write-a-function-log2x-which-gives-an-integer-approximation-of-the-log-base-2-of-a-positive-number-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/26\/write-a-function-log2x-which-gives-an-integer-approximation-of-the-log-base-2-of-a-positive-number-x\/","title":{"rendered":"Write a function log2(x), which gives an integer approximation of the log base 2 of a positive number x"},"content":{"rendered":"\n

Write a function log2(x), which gives an integer approximation of the log base 2 of a positive number x, but it does so using recursion. Use a base case where you return 0 if x is 1 or less. In the general case, add 1 to the answer and divide x by 2. On top of writing the function, also write a test case for the function to prove the program works.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To compute the integer approximation of the base-2 logarithm of a positive number x<\/code> using recursion, we can define a function log2(x)<\/code> that follows these steps:<\/p>\n\n\n\n

    \n
  1. Base Case<\/strong>: If x<\/code> is less than or equal to 1, return 0. This is because the logarithm of 1 is 0, and for any number less than 1, the logarithm is negative, but since we’re approximating to the nearest integer, we return 0.<\/li>\n\n\n\n
  2. Recursive Case<\/strong>: If x<\/code> is greater than 1, divide x<\/code> by 2 and add 1 to the result of the recursive call. This effectively counts how many times x<\/code> can be halved before it becomes less than or equal to 1, which corresponds to the integer part of the base-2 logarithm.<\/li>\n<\/ol>\n\n\n\n

    Here’s the Python implementation of the log2<\/code> function:<\/p>\n\n\n\n

    def log2(x):\n    # Base case: if x is 1 or less, return 0\n    if x <= 1:\n        return 0\n    # Recursive case: divide x by 2 and add 1\n    else:\n        return 1 + log2(x \/ 2)<\/code><\/pre>\n\n\n\n
    Test Case<\/strong>:<\/p>\n\n\n\n
    To verify that the function works correctly, we can test it with a known value. For example, the base-2 logarithm of 8 is 3, since 8 = 2\u00b3. Let’s test the function with x = 8<\/code>:<\/p>\n\n\n\n
    print(log2(8))  # Expected output: 3<\/code><\/pre>\n\n\n\n
    Explanation<\/strong>:<\/p>\n\n\n\n
      \n
    • Base Case<\/strong>: When x<\/code> is 1 or less, the function returns 0. This is because the logarithm of 1 is 0, and for any number less than 1, the logarithm is negative, but since we’re approximating to the nearest integer, we return 0.<\/li>\n\n\n\n
    • Recursive Case<\/strong>: For numbers greater than 1, the function recursively divides x<\/code> by 2 and adds 1. This process effectively counts how many times x<\/code> can be halved before it becomes less than or equal to 1, which corresponds to the integer part of the base-2 logarithm.<\/li>\n<\/ul>\n\n\n\n
      For example, with x = 8<\/code>:<\/p>\n\n\n\n
        \n
      1. log2(8)<\/code> calls log2(4)<\/code> and adds 1.<\/li>\n\n\n\n
      2. log2(4)<\/code> calls log2(2)<\/code> and adds 1.<\/li>\n\n\n\n
      3. log2(2)<\/code> calls log2(1)<\/code> and adds 1.<\/li>\n\n\n\n
      4. log2(1)<\/code> returns 0.<\/li>\n<\/ol>\n\n\n\n
        Adding up the results: 1 + 1 + 1 + 0 = 3, which is the correct integer approximation of the base-2 logarithm of 8.<\/p>\n\n\n\n
        This recursive approach effectively computes the integer part of the base-2 logarithm by counting how many times the number can be halved before it becomes less than or equal to 1.<\/p>\n","protected":false},"excerpt":{"rendered":"
        Write a function log2(x), which gives an integer approximation of the log base 2 of a positive number x, but it does so using recursion. Use a base case where you return 0 if x is 1 or less. In the general case, add 1 to the answer and divide x by 2. On top […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-186808","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186808","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=186808"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186808\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186808"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186808"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}