Solution 3.3:6b

From Förberedande kurs i matematik 1

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Current revision (07:59, 2 October 2008) (edit) (undo)
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The logarithm
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The logarithm <math>\lg 46</math> satisfies the relation
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<math>\text{lg 46 }</math>
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satisfies the relation
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<math>\text{10}^{\text{lg 46 }}=46</math>
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{{Displayed math||<math>10^{\lg 46} = 46</math>}}
and taking the natural logarithm of both sides, we obtain
and taking the natural logarithm of both sides, we obtain
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{{Displayed math||<math>\ln 10^{\lg 46 } = \ln 46\,\textrm{.}</math>}}
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<math>\ln \text{10}^{\text{lg 46 }}=\ln 46</math>
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If we use the logarithm law, <math>\lg a^b = b\cdot\lg a</math>, on the left-hand side, the equality becomes
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If we use the logarithm law,
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<math>\lg a^{b}=b\centerdot \lg a</math>, on the left-hand side, the equality becomes
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<math>\lg 46\centerdot \ln 10=\ln 46</math>
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{{Displayed math||<math>\lg 46\cdot\ln 10 = \ln 46\,\textrm{.}</math>}}
This shows that
This shows that
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{{Displayed math||<math>\lg 46 = \frac{\ln 46}{\ln 10} = \frac{3\textrm{.}828641\,\ldots}{2\textrm{.}302585\,\ldots} = 1\textrm{.}6627578\,\ldots</math>}}
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<math>\lg 46=\frac{\ln 46}{\ln 10}=\frac{3.828641}{2.302585}=1.6627578</math>
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and the answer is 1.663.
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and the answer is
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<math>\text{1}.\text{663}</math>.
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NOTE: In order to calculate the answer on a calculator, you press
 
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Note: In order to calculate the answer on the calculator, you press
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<math>\begin{align}
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<center>
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& \left[ 4 \right]\quad \left[ 6 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 1 \right]\quad \left[ 0 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\
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{|
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& \quad \\
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||
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\end{align}</math>
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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"|6
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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"|1
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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"|0
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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 logarithm \displaystyle \lg 46 satisfies the relation

\displaystyle 10^{\lg 46} = 46

and taking the natural logarithm of both sides, we obtain

\displaystyle \ln 10^{\lg 46 } = \ln 46\,\textrm{.}

If we use the logarithm law, \displaystyle \lg a^b = b\cdot\lg a, on the left-hand side, the equality becomes

\displaystyle \lg 46\cdot\ln 10 = \ln 46\,\textrm{.}

This shows that

\displaystyle \lg 46 = \frac{\ln 46}{\ln 10} = \frac{3\textrm{.}828641\,\ldots}{2\textrm{.}302585\,\ldots} = 1\textrm{.}6627578\,\ldots

and the answer is 1.663.


Note: In order to calculate the answer on the calculator, you press

4
  
6
  
LN
  
÷
  
1
  
0
  
LN
  
=
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