{"id":11318,"date":"2015-07-14T19:41:12","date_gmt":"2015-07-14T19:41:12","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11318"},"modified":"2015-09-09T19:13:41","modified_gmt":"2015-09-09T19:13:41","slug":"use-natural-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/use-natural-logarithms\/","title":{"raw":"Use natural logarithms","rendered":"Use natural logarithms"},"content":{"raw":"

The most frequently used base for logarithms is e<\/em>. Base e<\/em>\u00a0logarithms are important in calculus and some scientific applications; they are called natural logarithms<\/strong>. The base e<\/em>\u00a0logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].<\/p>\r\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 1 is always 0 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 e<\/em>\u00a0using the inverse property of logarithms.<\/p>\r\n\r\n

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A General Note: Definition of the Natural Logarithm<\/h3>\r\n

A natural logarithm<\/strong> is a logarithm with base e<\/em>. 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 x<\/em>\u00a0satisfies the following definition.<\/p>\r\n

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

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

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

The logarithm y<\/em>\u00a0is the exponent to which e<\/em>\u00a0must be raised to get x<\/em>.<\/p>\r\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 x<\/em>\u00a0and [latex]e^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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How To: Given a natural logarithm with the form [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex], evaluate it using a calculator.<\/h3>\r\n
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  1. Press [LN]<\/strong>.<\/li>\r\n\t
  2. Enter the value given for x<\/em>, followed by [ ) ]<\/strong>.<\/li>\r\n\t
  3. Press [ENTER]<\/strong>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
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    Example 6: Evaluating a Natural Logarithm Using a Calculator<\/h3>\r\n

    Evaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.<\/p>\r\n\r\n<\/div>\r\n

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    Solution<\/h3>\r\n