Solution 3.3:6a

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m (Lösning 3.3:6a moved to Solution 3.3:6a: Robot: moved page)
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{{NAVCONTENT_START}}
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The calculator does not have button for
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<center> [[Image:3_3_6a-1(2).gif]] </center>
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<math>\log _{3}</math>, but it does have one for the natural logarithm ln, so we need to rewrite
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{{NAVCONTENT_STOP}}
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<math>\log _{3}4</math>
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{{NAVCONTENT_START}}
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in terms of ln.
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<center> [[Image:3_3_6a-2(2).gif]] </center>
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{{NAVCONTENT_STOP}}
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If we go back to the definition of the logarithm, we see that
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[[Image:3_3_6_a.gif|center]]
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<math>\log _{3}4</math>
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is that number which satisfies
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<math>3^{\log _{3}4}=4</math>
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Now, take the natural logarithm of both sides,
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<math>\ln 3^{\log _{3}4}=\ln 4</math>
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Using the logarithm law,
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<math>\lg a^{b}=b\lg a</math>, the left-hand side can be written as
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<math>\log _{3}4\centerdot \ln 3</math>
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and the relation is
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<math>\log _{3}4\centerdot \ln 3=\ln 4</math>
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Thus, after dividing by
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<math>\text{ln 3}</math>, we have
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<math>\log _{3}4=\frac{\ln 4}{\ln 3}=\frac{1.386294...}{1.098612...}=1.2618595</math>
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which gives 1.262 as the rounded-off answer.
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NOTE: on a calculator, the answer is obtained by pressing the buttons
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<math>\begin{align}
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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>

Revision as of 09:08, 26 September 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 \begin{align} & \left[ 4 \right]\quad \left[ \text{LN} \right]\quad \left[ \div \right]\quad \left[ 3 \right]\quad \left[ \text{LN} \right]\quad \left[ = \right] \\ & \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] \\ \end{align}

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