Lösung 3.3:6a

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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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
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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{{Displayed math||<math>3^{\log _{3}4} = 4\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<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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{{Abgesetzte Formel||<math>\ln 3^{\log _{3}4}=\ln 4\,\textrm{.}</math>}}
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
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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{{Displayed math||<math>\log_{3}4\cdot \ln 3 = \ln 4\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>\log_{3}4\cdot \ln 3 = \ln 4\,\textrm{.}</math>}}
Thus, after dividing by <math>\ln 3</math>, we have
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>}}
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{{Abgesetzte Formel||<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.

Version vom 08:45, 22. Okt. 2008

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
  
=
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_3.3:6a“