Lösung 3.3:6c
Aus Online Mathematik Brückenkurs 1
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Before we even start thinking about transforming <math>\log_2</math> and <math>\log_3</math> to ln, we use the log laws | Before we even start thinking about transforming <math>\log_2</math> and <math>\log_3</math> to ln, we use the log laws | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\log a^b &= b\cdot\log a\,,\\[5pt] | \log a^b &= b\cdot\log a\,,\\[5pt] | ||
\log (a\cdot b) &= \log a+\log b\,, | \log (a\cdot b) &= \log a+\log b\,, | ||
| Zeile 8: | Zeile 8: | ||
to simplify the expression | to simplify the expression | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\log_{3}\log _{2}3^{118} | \log_{3}\log _{2}3^{118} | ||
&= \log_{3}(118\cdot\log_{2}3)\\[5pt] | &= \log_{3}(118\cdot\log_{2}3)\\[5pt] | ||
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With help of the relation <math>2^{\log_{2}x} = x</math> and <math>3^{\log_{3}x} = x</math> and taking the natural logarithm , we can express <math>\log_{2}</math> and <math>\log_{3}</math> using ln, | With help of the relation <math>2^{\log_{2}x} = x</math> and <math>3^{\log_{3}x} = x</math> and taking the natural logarithm , we can express <math>\log_{2}</math> and <math>\log_{3}</math> using ln, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\log_{2}x=\frac{\ln x}{\ln 2}\quad</math> and <math>\quad\log_{3}x = \frac{\ln x}{\ln 3}\,\textrm{.}</math>}} |
The two terms <math>\log_3 118</math> and <math>\log_3\log_2 3</math> can therefore be written as | The two terms <math>\log_3 118</math> and <math>\log_3\log_2 3</math> can therefore be written as | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\log_{3}118 = \frac{\ln 118}{\ln 3}\quad</math> and <math>\quad\log_{3}\log_{2}3 = \log_{3}\frac{\ln 3}{\ln 2}\,,</math>}} |
where we can simplify the last expression further with the logarithm law, log (a/b) = log a – log b, and then transform <math>\log _{3}</math> to ln, | where we can simplify the last expression further with the logarithm law, log (a/b) = log a – log b, and then transform <math>\log _{3}</math> to ln, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\log_{3}\frac{\ln 3}{\ln 2} | \log_{3}\frac{\ln 3}{\ln 2} | ||
&= \log_{3}\ln 3 - \log_{3}\ln 2\\[5pt] | &= \log_{3}\ln 3 - \log_{3}\ln 2\\[5pt] | ||
| Zeile 32: | Zeile 32: | ||
In all, we thus obtain | In all, we thus obtain | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\log_{3}\log_{2}3^{118} = \frac{\ln 118}{\ln 3} + \frac{\ln \ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.}</math>}} |
Input into the calculator gives | Input into the calculator gives | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\log_{3}\log_{2}3^{118}\approx 4\textrm{.}762\,\textrm{.}</math>}} |
Version vom 08:45, 22. Okt. 2008
Before we even start thinking about transforming \displaystyle \log_2 and \displaystyle \log_3 to ln, we use the log laws
| \displaystyle \begin{align}
\log a^b &= b\cdot\log a\,,\\[5pt] \log (a\cdot b) &= \log a+\log b\,, \end{align} |
to simplify the expression
| \displaystyle \begin{align}
\log_{3}\log _{2}3^{118} &= \log_{3}(118\cdot\log_{2}3)\\[5pt] &= \log_{3}118 + \log_{3}\log_{2}3\,\textrm{.} \end{align} |
With help of the relation \displaystyle 2^{\log_{2}x} = x and \displaystyle 3^{\log_{3}x} = x and taking the natural logarithm , we can express \displaystyle \log_{2} and \displaystyle \log_{3} using ln,
| \displaystyle \log_{2}x=\frac{\ln x}{\ln 2}\quad and \displaystyle \quad\log_{3}x = \frac{\ln x}{\ln 3}\,\textrm{.} |
The two terms \displaystyle \log_3 118 and \displaystyle \log_3\log_2 3 can therefore be written as
| \displaystyle \log_{3}118 = \frac{\ln 118}{\ln 3}\quad and \displaystyle \quad\log_{3}\log_{2}3 = \log_{3}\frac{\ln 3}{\ln 2}\,, |
where we can simplify the last expression further with the logarithm law, log (a/b) = log a – log b, and then transform \displaystyle \log _{3} to ln,
| \displaystyle \begin{align}
\log_{3}\frac{\ln 3}{\ln 2} &= \log_{3}\ln 3 - \log_{3}\ln 2\\[5pt] &= \frac{\ln\ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.} \end{align} |
In all, we thus obtain
| \displaystyle \log_{3}\log_{2}3^{118} = \frac{\ln 118}{\ln 3} + \frac{\ln \ln 3}{\ln 3} - \frac{\ln\ln 2}{\ln 3}\,\textrm{.} |
Input into the calculator gives
| \displaystyle \log_{3}\log_{2}3^{118}\approx 4\textrm{.}762\,\textrm{.} |
Note: The button sequence on the calculator will be:
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