Lösung 3.3:6a

Aus Online Mathematik Brückenkurs 1

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<math>\begin{align}
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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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& \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\
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& \quad \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\
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\end{align}</math>
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Version vom 09:25, 26. Sep. 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


Now, take the natural logarithm of both sides,


\displaystyle \ln 3^{\log _{3}4}=\ln 4


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


\displaystyle \log _{3}4\centerdot \ln 3=\ln 4


Thus, after dividing by \displaystyle \text{ln 3}, we have


\displaystyle \log _{3}4=\frac{\ln 4}{\ln 3}=\frac{1.386294...}{1.098612...}=1.2618595


which gives 1.262 as the rounded-off answer.

NOTE: on a calculator, the answer is obtained by pressing the buttons


\displaystyle \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right]

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