{"id":11316,"date":"2015-07-14T19:40:56","date_gmt":"2015-07-14T19:40:56","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11316"},"modified":"2021-11-07T17:11:05","modified_gmt":"2021-11-07T17:11:05","slug":"use-common-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/use-common-logarithms\/","title":{"raw":"Use common logarithms","rendered":"Use common logarithms"},"content":{"raw":"

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression [latex]\\mathrm{log}\\left(x\\right)[\/latex] means [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex]. We call a base-10 logarithm a common logarithm<\/strong>. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.<\/p>\r\n\r\n

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Definition of the Common Logarithm<\/div>\r\n

A common logarithm<\/span> is a logarithm with base [latex]10[\/latex]. We write [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{log}\\left(x\\right)[\/latex]. The common logarithm of a positive number [latex]x[\/latex] satisfies the following definition.<\/p>\r\n

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

[latex]y=\\mathrm{log}\\left(x\\right)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/div>\r\n

We read [latex]\\mathrm{log}\\left(x\\right)[\/latex] as, \"the logarithm with base [latex]10[\/latex] of [latex]x[\/latex] \" or \"log base 10 of [latex]x[\/latex]. \"<\/p>\r\n

The logarithm [latex]y[\/latex] is the exponent to which [latex]10[\/latex] must be raised to get [latex]x[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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Given a common logarithm of the form [latex]y=\\mathrm{log}\\left(x\\right)[\/latex], evaluate it mentally.<\/strong><\/p>\r\n\r\n

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  1. Rewrite the argument [latex]x[\/latex] as a power of [latex]10:[\/latex] [latex]{10}^{y}=x[\/latex].<\/li>\r\n \t
  2. Use previous knowledge of powers of [latex]10[\/latex] to identify [latex]y[\/latex] by asking, \"To what exponent must [latex]10[\/latex] be raised in order to get [latex]x?[\/latex] \"<\/li>\r\n<\/ol>\r\n<\/div>\r\n
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    Finding the Value of a Common Logarithm Mentally<\/div>\r\n

    Evaluate [latex]y=\\mathrm{log}\\left(1000\\right)[\/latex] without using a calculator.<\/p>\r\n\r\n<\/div>\r\n

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    First we rewrite the logarithm in exponential form: [latex]{10}^{y}=1000[\/latex]. Next, we ask, \"To what exponent must [latex]10[\/latex] be raised in order to get 1000?\" We know<\/p>\r\n\r\n

    [latex]{10}^{3}=1000[\/latex]<\/div>\r\n

    Therefore, [latex]\\mathrm{log}\\left(1000\\right)=3[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Evaluate [latex]y=\\mathrm{log}\\left(1,000,000\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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    [latex]\\mathrm{log}\\left(1,000,000\\right)=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Given a common logarithm with the form [latex]y=\\mathrm{log}\\left(x\\right)[\/latex], evaluate it using a calculator.<\/strong><\/p>\r\n\r\n

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    1. Press [LOG]<\/strong>.<\/li>\r\n \t
    2. Enter the value given for [latex]x[\/latex], followed by [ ) ]<\/strong>.<\/li>\r\n \t
    3. Press [ENTER]<\/strong>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
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      Finding the Value of a Common Logarithm Using a Calculator<\/div>\r\n

      Evaluate [latex]y=\\mathrm{log}\\left(321\\right)[\/latex] to four decimal places using a calculator.<\/p>\r\n\r\n<\/div>\r\n

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