Solution 3.3:6a
From Förberedande kurs i matematik 1
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| - | {{ | + | The calculator does not have button for <math>\log_{3}</math>, but it does have one for the natural logarithm ln, so we need to rewrite <math>\log_{3}4</math> in terms of ln. |
| - | < | + | |
| - | {{ | + | 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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| - | + | 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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| + | 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>}} | ||
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| + | 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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| + | which gives 1.262 as the rounded-off answer. | ||
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| + | Note: On the calculator, the answer is obtained by pressing the buttons | ||
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| + | <center> | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|4 | ||
| + | |} | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|LN | ||
| + | |} | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|÷ | ||
| + | |} | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|3 | ||
| + | |} | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|LN | ||
| + | |} | ||
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| + | {| border="1" cellpadding="3" cellspacing="0" | ||
| + | |width="30px" align="center"|= | ||
| + | |} | ||
| + | |} | ||
| + | </center> | ||
Current revision
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
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