{"id":1062,"date":"2025-07-22T00:07:16","date_gmt":"2025-07-22T00:07:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1062"},"modified":"2026-03-17T18:17:03","modified_gmt":"2026-03-17T18:17:03","slug":"logarithmic-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-functions-learn-it-4\/","title":{"raw":"Logarithmic Functions: Learn It 4","rendered":"Logarithmic Functions: Learn It 4"},"content":{"raw":"

Natural Logarithms<\/h2>\r\nThe most frequently used base for logarithms is [latex]e[\/latex]. Base [latex]e[\/latex]\u00a0logarithms are important in calculus and some scientific applications; they are called natural logarithms<\/strong>. The base [latex]e[\/latex]\u00a0logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].\r\n\r\nMost values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of [latex]1[\/latex] is always [latex]0[\/latex] in any base, [latex]\\mathrm{ln}(1)=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of [latex]e[\/latex] using the inverse property of logarithms.\r\n\r\n
\r\n

natural logarithm<\/h3>\r\nA natural logarithm<\/strong> is a logarithm with base [latex]e[\/latex]. We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]. The natural logarithm of a positive number [latex]x[\/latex]\u00a0satisfies the following definition:\r\n\r\nFor [latex]x>0[\/latex],\r\n

[latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/p>\r\n \r\n\r\nWe read [latex]\\mathrm{ln}\\left(x\\right)[\/latex] as, \"the logarithm with base [latex]e[\/latex]\u00a0of [latex]x[\/latex]\" or \"the natural logarithm of [latex]x[\/latex].\"\r\n

    \r\n \t
  • The logarithm [latex]y[\/latex]\u00a0is the exponent to which [latex]e[\/latex]\u00a0must be raised to get [latex]x[\/latex].<\/li>\r\n \t
  • Since the functions [latex]y=e^{x}[\/latex] and [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{ln}\\left({e}^{x}\\right)=x[\/latex] for all [latex]x[\/latex]\u00a0and [latex]e^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
    Evaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.[reveal-answer q=\"810207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"810207\"]\r\n
      \r\n \t
    • Press [LN]<\/strong>.<\/li>\r\n \t
    • Enter [latex]500[\/latex], followed by [ ) ]<\/strong>.<\/li>\r\n \t
    • Press [ENTER]<\/strong>.<\/li>\r\n<\/ul>\r\nRounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      Evaluate [latex]\\mathrm{ln}\\left(-500\\right)[\/latex].[reveal-answer q=\"342695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"342695\"]It is not possible to take the logarithm of a negative number in the set of real numbers.[\/hidden-answer]<\/section>
      [ohm_question hide_question_numbers=1]321433[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321434[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321435[\/ohm_question]<\/section>","rendered":"

      Natural Logarithms<\/h2>\n

      The most frequently used base for logarithms is [latex]e[\/latex]. Base [latex]e[\/latex]\u00a0logarithms are important in calculus and some scientific applications; they are called natural logarithms<\/strong>. The base [latex]e[\/latex]\u00a0logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].<\/p>\n

      Most values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of [latex]1[\/latex] is always [latex]0[\/latex] in any base, [latex]\\mathrm{ln}(1)=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of [latex]e[\/latex] using the inverse property of logarithms.<\/p>\n

      \n

      natural logarithm<\/h3>\n

      A natural logarithm<\/strong> is a logarithm with base [latex]e[\/latex]. We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]. The natural logarithm of a positive number [latex]x[\/latex]\u00a0satisfies the following definition:<\/p>\n

      For [latex]x>0[\/latex],<\/p>\n

      [latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/p>\n

       <\/p>\n

      We read [latex]\\mathrm{ln}\\left(x\\right)[\/latex] as, “the logarithm with base [latex]e[\/latex]\u00a0of [latex]x[\/latex]” or “the natural logarithm of [latex]x[\/latex].”<\/p>\n

        \n
      • The logarithm [latex]y[\/latex]\u00a0is the exponent to which [latex]e[\/latex]\u00a0must be raised to get [latex]x[\/latex].<\/li>\n
      • Since the functions [latex]y=e^{x}[\/latex] and [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{ln}\\left({e}^{x}\\right)=x[\/latex] for all [latex]x[\/latex]\u00a0and [latex]e^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/li>\n<\/ul>\n<\/section>\n
        Evaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.<\/p>\n