Lösung 3.4:2a

Aus Online Mathematik Brückenkurs 1

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

Version vom 10:11, 26. Sep. 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.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_3.4:2a“