Solution 3.3:6a

From Förberedande kurs i matematik 1

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Current revision (07:53, 2 October 2008) (edit) (undo)
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The calculator does not have button for
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The calculator does not have button for <math>\log_{3}</math>, but it does have one for the natural logarithm ln, so we need to rewrite <math>\log_{3}4</math> in terms of ln.
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<math>\log _{3}</math>, but it does have one for the natural logarithm ln, so we need to rewrite
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<math>\log _{3}4</math>
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in terms of ln.
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If we go back to the definition of the logarithm, we see that
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If we go back to the definition of the logarithm, we see that <math>\log _{3}4</math> is that number which satisfies
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<math>\log _{3}4</math>
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is that number which satisfies
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<math>3^{\log _{3}4}=4</math>
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{{Displayed math||<math>3^{\log _{3}4} = 4\,\textrm{.}</math>}}
Now, take the natural logarithm of both sides,
Now, take the natural logarithm of both sides,
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{{Displayed math||<math>\ln 3^{\log _{3}4}=\ln 4\,\textrm{.}</math>}}
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<math>\ln 3^{\log _{3}4}=\ln 4</math>
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Using the logarithm law, <math>\lg a^b = b\lg a</math>, the left-hand side can be written as <math>\log_{3}4\cdot\ln 3</math> and the relation is
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Using the logarithm law,
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<math>\lg a^{b}=b\lg a</math>, the left-hand side can be written as
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<math>\log _{3}4\centerdot \ln 3</math>
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and the relation is
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<math>\log _{3}4\centerdot \ln 3=\ln 4</math>
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Thus, after dividing by
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<math>\text{ln 3}</math>, we have
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{{Displayed math||<math>\log_{3}4\cdot \ln 3 = \ln 4\,\textrm{.}</math>}}
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<math>\log _{3}4=\frac{\ln 4}{\ln 3}=\frac{1.386294...}{1.098612...}=1.2618595</math>
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Thus, after dividing by <math>\ln 3</math>, we have
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{{Displayed math||<math>\log_{3}4 = \frac{\ln 4}{\ln 3} = \frac{1\textrm{.}386294\,\ldots}{1\textrm{.}098612\,\ldots} = 1\textrm{.}2618595\,\ldots</math>}}
which gives 1.262 as the rounded-off answer.
which gives 1.262 as the rounded-off answer.
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NOTE: on a calculator, the answer is obtained by pressing the buttons
 
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Note: On the calculator, the answer is obtained by pressing the buttons
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<math>\left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right]</math>
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<center>
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{|
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|4
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|}
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||&nbsp;&nbsp;
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|LN
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|}
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||&nbsp;&nbsp;
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|÷
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|}
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||&nbsp;&nbsp;
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|3
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|}
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||&nbsp;&nbsp;
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|LN
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|}
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||&nbsp;&nbsp;
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||
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{| border="1" cellpadding="3" cellspacing="0"
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|width="30px" align="center"|=
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|}
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|}
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</center>

Current revision

The calculator does not have button for \displaystyle \log_{3}, but it does have one for the natural logarithm ln, so we need to rewrite \displaystyle \log_{3}4 in terms of ln.

If we go back to the definition of the logarithm, we see that \displaystyle \log _{3}4 is that number which satisfies

\displaystyle 3^{\log _{3}4} = 4\,\textrm{.}

Now, take the natural logarithm of both sides,

\displaystyle \ln 3^{\log _{3}4}=\ln 4\,\textrm{.}

Using the logarithm law, \displaystyle \lg a^b = b\lg a, the left-hand side can be written as \displaystyle \log_{3}4\cdot\ln 3 and the relation is

\displaystyle \log_{3}4\cdot \ln 3 = \ln 4\,\textrm{.}

Thus, after dividing by \displaystyle \ln 3, we have

\displaystyle \log_{3}4 = \frac{\ln 4}{\ln 3} = \frac{1\textrm{.}386294\,\ldots}{1\textrm{.}098612\,\ldots} = 1\textrm{.}2618595\,\ldots

which gives 1.262 as the rounded-off answer.


Note: On the calculator, the answer is obtained by pressing the buttons

4
  
LN
  
÷
  
3
  
LN
  
=
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