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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 log3, but it does have one for the natural logarithm ln, so we need to rewrite log34 in terms of ln.

If we go back to the definition of the logarithm, we see that log34 is that number which satisfies

3log34=4.

Now, take the natural logarithm of both sides,

ln3log34=ln4.

Using the logarithm law, lgab=blga, the left-hand side can be written as log34ln3 and the relation is

log34ln3=ln4.

Thus, after dividing by ln3, we have

log34=ln3ln4=1.098612...1.386294...=1.2618595...

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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