Solution 3.4:2a

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m (Lösning 3.4:2a moved to Solution 3.4:2a: Robot: moved page)
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The left-hand side is "
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<center> [[Image:3_4_2a.gif]] </center>
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<math>\text{2}</math>
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raised to something", and therefore a positive number regardless of whatever value the exponent has. We can therefore take the log of both sides,
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<math>\ln 2^{x^{2}-2}=\ln 1</math>
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and use the log law
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<math>\lg a^{b}=b\centerdot \lg a</math>
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to get the exponent
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<math>x^{\text{2}}-\text{2 }</math>
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as a factor on the left-hand side
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<math>\left( x^{\text{2}}-\text{2 } \right)\ln 2=\ln 1</math>
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Because
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<math>e^{0}=1</math>, so
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<math>\text{ln 1}=0</math>, giving:
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<math>\left( x^{\text{2}}-\text{2 } \right)\ln 2=0</math>
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This means that
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<math>x</math>
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must satisfy the second-degree equation
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<math>\left( x^{\text{2}}-\text{2 } \right)=0</math>
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Taking the root gives
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<math>x=-\sqrt{2}</math>
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or
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<math>x=\sqrt{2}.</math>
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NOTE: the exercise is taken from a Finnish upper-secondary final examination from March 2007.

Revision as of 10:11, 26 September 2008

The left-hand side is " \displaystyle \text{2} raised to something", and therefore a positive number regardless of whatever value the exponent has. We can therefore take the log of both sides,


\displaystyle \ln 2^{x^{2}-2}=\ln 1

and use the log law \displaystyle \lg a^{b}=b\centerdot \lg a to get the exponent \displaystyle x^{\text{2}}-\text{2 } as a factor on the left-hand side


\displaystyle \left( x^{\text{2}}-\text{2 } \right)\ln 2=\ln 1


Because \displaystyle e^{0}=1, so \displaystyle \text{ln 1}=0, giving:


\displaystyle \left( x^{\text{2}}-\text{2 } \right)\ln 2=0


This means that \displaystyle x must satisfy the second-degree equation


\displaystyle \left( x^{\text{2}}-\text{2 } \right)=0


Taking the root gives \displaystyle x=-\sqrt{2} or \displaystyle x=\sqrt{2}.


NOTE: the exercise is taken from a Finnish upper-secondary final examination from March 2007.

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