Aus Online Mathematik Brückenkurs 1
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
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| \displaystyle 3^{\log _{3}4} = 4\,\textrm{.}
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Now, take the natural logarithm of both sides,
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| \displaystyle \ln 3^{\log _{3}4}=\ln 4\,\textrm{.}
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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
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| \displaystyle \log_{3}4\cdot \ln 3 = \ln 4\,\textrm{.}
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Thus, after dividing by \displaystyle \ln 3, we have
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| \displaystyle \log_{3}4 = \frac{\ln 4}{\ln 3} = \frac{1\textrm{.}386294\,\ldots}{1\textrm{.}098612\,\ldots} = 1\textrm{.}2618595\,\ldots
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which gives 1.262 as the rounded-off answer.
Note: On the calculator, the answer is obtained by pressing the buttons
