Lösung 3.4:1a - Versionsgeschichte

Markus.bez: Sprache und Formulierung

2009-06-12T13:43:35Z

Sprache und Formulierung

← Nächstältere Version		Version vom 13:43, 12. Jun. 2009
Zeile 1:		Zeile 1:
-	Wir logarithmieren beide Seiten mit ~~den~~ natürlichen Logarithmus	+	Wir logarithmieren beide Seiten mit dem natürlichen Logarithmus:

	{{Abgesetzte Formel\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}

-	Danach verwenden wir das Logarithmusgesetz <math>\ln a^{b} = b\cdot \ln a</math>, und erhalten dadurch	+	Danach verwenden wir das Logarithmusgesetz <math>\ln a^{b} = b\cdot \ln a</math> und erhalten dadurch

-	{{Abgesetzte Formel\|\|<math>x\cdot \ln e = \ln 13</math>}}	+	{{Abgesetzte Formel\|\|<math>x\cdot \ln e = \ln 13\,.</math>}}

-	~~und~~ lösen die Gleichung für <math>x</math>,	+	Wir lösen die Gleichung für <math>x</math>:

	{{Abgesetzte Formel\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}


-	Hinweis: Eigentlich müssen wir sicherstellen ob beide Seiten positiv sind, bevor wir logarithmieren, ~~nachdem~~ der Logarithmus von negativen Zahlen nicht definiert ist. Nachdem wir die Gleichung	+	Hinweis: Eigentlich müssen wir sicherstellen, dass beide Seiten positiv sind, bevor wir logarithmieren, weil der Logarithmus von negativen Zahlen nicht definiert ist. Nachdem wir die Gleichung

	{{Abgesetzte Formel\|\|<math>e^x=13</math>}}		{{Abgesetzte Formel\|\|<math>e^x=13</math>}}

-	haben, sehen wir direkt dass die rechte Seite positiv ist. Die linke Seite ist auch positiv, nachdem jede Potenz von ''e'' positiv ist.	+	haben, sehen wir jedoch direkt, dass die rechte Seite positiv ist. Die linke Seite ist auch positiv, nachdem jede Potenz von ''e'' positiv ist.

Martin um 16:03, 27. Mär. 2009

2009-03-27T16:03:20Z

← Nächstältere Version		Version vom 16:03, 27. Mär. 2009
Zeile 1:		Zeile 1:
-	~~We solve an equation of this type by taking the natural logarithm of both sides~~	+	Wir logarithmieren beide Seiten mit den natürlichen Logarithmus

	{{Abgesetzte Formel\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}

-	~~Then, using the log law~~ <math>\ln a^{b} = b\cdot \ln a</math>, ~~the unknown <math>x</math> can then be moved down as a factor on the left-hand side~~	+	Danach verwenden wir das Logarithmusgesetz <math>\ln a^{b} = b\cdot \ln a</math>, und erhalten dadurch

	{{Abgesetzte Formel\|\|<math>x\cdot \ln e = \ln 13</math>}}		{{Abgesetzte Formel\|\|<math>x\cdot \ln e = \ln 13</math>}}

-	~~and then it is simple to solve for~~ <math>x</math>,	+	und lösen die Gleichung für <math>x</math>,

	{{Abgesetzte Formel\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}


-	~~Note~~: ~~One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation~~, ~~it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number)~~. ~~Because the equation is~~	+	Hinweis: Eigentlich müssen wir sicherstellen ob beide Seiten positiv sind, bevor wir logarithmieren, nachdem der Logarithmus von negativen Zahlen nicht definiert ist. Nachdem wir die Gleichung

	{{Abgesetzte Formel\|\|<math>e^x=13</math>}}		{{Abgesetzte Formel\|\|<math>e^x=13</math>}}

-	~~we see directly that the right-hand side is positive~~. ~~The left-hand side is also positive because “a positive number <math>~~e~~\approx 2\textrm{.}718</math> raised to a number” always gives a positive number~~.	+	haben, sehen wir direkt dass die rechte Seite positiv ist. Die linke Seite ist auch positiv, nachdem jede Potenz von ''e'' positiv ist.

Tekbot: hat „Solution 3.4:1a“ nach „Lösung 3.4:1a“ verschoben: Robot: moved page

2008-10-22T14:40:24Z

hat „Solution 3.4:1a“ nach „Lösung 3.4:1a“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 14:40, 22. Okt. 2008

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

2008-10-22T08:45:38Z

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

← Nächstältere Version		Version vom 08:45, 22. Okt. 2008
Zeile 1:		Zeile 1:
	We solve an equation of this type by taking the natural logarithm of both sides		We solve an equation of this type by taking the natural logarithm of both sides

-	{{~~Displayed math~~\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}

	Then, using the log law <math>\ln a^{b} = b\cdot \ln a</math>, the unknown <math>x</math> can then be moved down as a factor on the left-hand side		Then, using the log law <math>\ln a^{b} = b\cdot \ln a</math>, the unknown <math>x</math> can then be moved down as a factor on the left-hand side

-	{{~~Displayed math~~\|\|<math>x\cdot \ln e = \ln 13</math>}}	+	{{Abgesetzte Formel\|\|<math>x\cdot \ln e = \ln 13</math>}}

	and then it is simple to solve for <math>x</math>,		and then it is simple to solve for <math>x</math>,

-	{{~~Displayed math~~\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}


	Note: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is		Note: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is

-	{{~~Displayed math~~\|\|<math>e^x=13</math>}}	+	{{Abgesetzte Formel\|\|<math>e^x=13</math>}}

	we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number <math>e\approx 2\textrm{.}718</math> raised to a number” always gives a positive number.		we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number <math>e\approx 2\textrm{.}718</math> raised to a number” always gives a positive number.

Tek um 08:45, 2. Okt. 2008

2008-10-02T08:45:44Z

← Nächstältere Version		Version vom 08:45, 2. Okt. 2008
Zeile 1:		Zeile 1:
-	We solve an equation of this type by taking the natural logarithm of both sides:	+	We solve an equation of this type by taking the natural logarithm of both sides

		+	{{Displayed math\|\|<math>\ln e^x = \ln 13\,\textrm{.}</math>}}

-	<math>\ln e^{x}=\ln 13</math>	+	Then, using the log law <math>\ln a^{b} = b\cdot \ln a</math>, the unknown <math>x</math> can then be moved down as a factor on the left-hand side

		+	{{Displayed math\|\|<math>x\cdot \ln e = \ln 13</math>}}

-	~~Then, using the log law~~	+	and then it is simple to solve for <math>x</math>,
-	<math>~~\ln a^{b}=b\centerdot \ln a~~</math>, ~~the unknown~~	+
-	~~<math>x</math>~~	+
-	~~can then be moved down as a factor on the left-hand side~~	+

		+	{{Displayed math\|\|<math>x = \frac{\ln 13}{\ln e} = \frac{\ln 13}{1} = \ln 13\,\textrm{.}</math>}}

-	<math>x\centerdot \ln e=\ln 13</math>

		+	Note: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is

-	~~and then it is simple to solve for~~	+	{{Displayed math\|\|<math>e^x=13</math>}}
-	<math>x</math>	+
-	,	+

-		+	we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number <math>e\approx 2\textrm{.}718</math> raised to a number” always gives a positive number.
-	~~<math>x=\frac{\ln 13}{\ln e}=\frac{\ln 13}{1}=\ln 13</math>~~	+
-		+
-		+
-	NOTE: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is	+
-		+
-		+
-	~~<math>e^{x}=13</math>~~	+
-		+
-		+
-	we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number	+
-	<math>e\approx 2.718</math>	+
-	raised to a number” always gives a positive number.	+

Ian um 12:22, 12. Sep. 2008

2008-09-12T12:22:20Z

← Nächstältere Version		Version vom 12:22, 12. Sep. 2008
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-	{{~~NAVCONTENT_START}~~}	+	We solve an equation of this type by taking the natural logarithm of both sides:
-	<~~center~~> ~~[[Image:3_4_1a~~-~~1(2).gif]]~~ </~~center~~>	+
-	{{~~NAVCONTENT_STOP~~}}	+
-	{~~{NAVCONTENT_START~~}}	+	<math>\ln e^{x}=\ln 13</math>
-	<~~center~~> ~~[[Image~~:~~3_4_1a~~-2(2).~~gif]]~~ <~~/center~~>	+
-	{~~{NAVCONTENT_STOP}~~}	+
		+	Then, using the log law
		+	<math>\ln a^{b}=b\centerdot \ln a</math>, the unknown
		+	<math>x</math>
		+	can then be moved down as a factor on the left-hand side
		+
		+
		+	<math>x\centerdot \ln e=\ln 13</math>
		+
		+
		+	and then it is simple to solve for
		+	<math>x</math>
		+	,
		+
		+
		+	<math>x=\frac{\ln 13}{\ln e}=\frac{\ln 13}{1}=\ln 13</math>
		+
		+
		+	NOTE: One thing that we haven't mentioned in this solution is that before we carry out the first step and take the logarithm of both sides of the equation, it is necessary to be certain that the left- and right-hand sides are positive (it is not possible to take the logarithm of a negative number). Because the equation is
		+
		+
		+	<math>e^{x}=13</math>
		+
		+
		+	we see directly that the right-hand side is positive. The left-hand side is also positive because “a positive number
		+	<math>e\approx 2.718</math>
		+	raised to a number” always gives a positive number.

Tekbot: Lösning 3.4:1a moved to Solution 3.4:1a: Robot: moved page

2008-09-09T09:30:03Z

Lösning 3.4:1a moved to Solution 3.4:1a: Robot: moved page

← Nächstältere Version

Version vom 09:30, 9. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

2008-08-21T07:02:09Z

Robot: Automated text replacement (-[[Bild: +[[Image:)

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2008-04-02T10:27:05Z

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