3.3 Logarithms - Revision history

Phil at 16:48, 2 January 2009

Phil — Fri, 02 Jan 2009 16:48:13 GMT

←Older revision		Revision as of 16:48, 2 January 2009
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	{{Displayed math\|\|<math>10^{\ \bbox[#AAEEFF,2pt]{\,x\,}} = y\,\mbox{.} </math>}}		{{Displayed math\|\|<math>10^{\ \bbox[#AAEEFF,2pt]{\,x\,}} = y\,\mbox{.} </math>}}

-	One way to think about this is that the logarithm of <math>y</math> is the answer to the question "10 to the power ''what'' is equal to <math>y</math>?" Thus, <math>\lg 100</math> is the answer to the question "10 to the power ''what'' is equal to <math>1000</math>?", this answer is clearly 3.	+	One way to think about this is that the logarithm of <math>y</math> is the answer to the question "10 to the power ''what'' is equal to <math>y</math>?" Thus, <math>\lg 100</math> is the answer to the question "10 to the power ''what'' is equal to <math>1000</math>?"; this answer is clearly 3.

	Note that <math>y</math> must be a positive number for the logarithm <math>\lg y</math> to be defined, since there is no power of 10 that evaluates to a negative number or zero .		Note that <math>y</math> must be a positive number for the logarithm <math>\lg y</math> to be defined, since there is no power of 10 that evaluates to a negative number or zero .

Phil: Edited for UK terminology, notation and idiom. Informal explanation of logarithms added: "10 to the power what?" Delete if you don't like it.

Phil — Fri, 02 Jan 2009 16:40:11 GMT

Edited for UK terminology, notation and idiom. Informal explanation of logarithms added: "10 to the power what?" Delete if you don't like it.

(Difference between revisions)

Jdinglis at 22:30, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 22:30:16 GMT

←Older revision		Revision as of 22:30, 28 October 2008
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	You may need to spend some time studying logarithms.		You may need to spend some time studying logarithms.

-	Logarithms ~~usually~~ are not usually dealt with in detail in high school. Therefore, many college students tend to encounter problems when it comes to calculations with logarithms.	+	Logarithms are not usually dealt with in detail in high school. Therefore, many college students tend to encounter problems when it comes to calculations with logarithms.

Jdinglis at 20:44, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:44:47 GMT

←Older revision		Revision as of 20:44, 28 October 2008
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	</div>		</div>

-	The general formula for changing from one base <math>a</math> to another base <math>b</math> ~~can be derived in the same way~~	+	The general formula for changing from one base <math>a</math> to another base <math>b</math> is

	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\log_{\scriptstyle\,a} x		\log_{\scriptstyle\,a} x
	= \frac{\log_{\scriptstyle\, b} x}{\log_{\scriptstyle\, b} a}		= \frac{\log_{\scriptstyle\, b} x}{\log_{\scriptstyle\, b} a}
-	\,\mbox{.}</math>}}	+	\,\mbox{,}</math>}}

-	~~If one wants to~~ change the base of a power~~, one can do this by~~ using logarithms. For instance, if we want to write <math> 2^5 </math> using the base 10 ~~one~~ first ~~writes~~ 2 as a power with the base 10;	+	and can be derived in the same way. We can also change the base of a power using logarithms. For instance, if we want to write <math> 2^5 </math> using the base 10 first write 2 as a power with the base 10:
-	{{Displayed math\|\|<math>2 = 10^{\lg 2}</math>}}	+	{{Displayed math\|\|<math>2 = 10^{\lg 2}</math>.}}

-	~~and then~~ using one of the laws of exponents	+	Then, using one of the laws of exponents,
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	2^5 = (10^{\lg 2})^5 = 10^{5\cdot \lg 2}		2^5 = (10^{\lg 2})^5 = 10^{5\cdot \lg 2}
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	<br>		<br>
	First, we write 10 as a power of ''e'',		First, we write 10 as a power of ''e'',
-	{{Displayed math\|\|<math>10 = e^{\ln 10}</math>}}	+	{{Displayed math\|\|<math>10 = e^{\ln 10}</math>.}}

-	~~and then use~~ the laws of exponents	+	Using the the laws of exponents we can then see that
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	10^x = (e^{\ln 10})^x = e^{\,x \cdot \ln 10}		10^x = (e^{\ln 10})^x = e^{\,x \cdot \ln 10}
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-	'''Keep in mind that: '''	+	'''Keep in mind that... '''

-	You may need to spend ~~much~~ time studying logarithms.	+	You may need to spend some time studying logarithms.

-	Logarithms usually are dealt with ~~summarily~~ in high school. Therefore, many college students tend to encounter problems when it comes to calculations with logarithms.	+	Logarithms usually are not usually dealt with in detail in high school. Therefore, many college students tend to encounter problems when it comes to calculations with logarithms.


	'''Reviews'''		'''Reviews'''

-	For those of you who want to deepen your ~~studies~~ or need more detailed explanations consider the following references:	+	For those of you who want to deepen your understanding or need more detailed explanations consider the following references:

-	[http://en.wikipedia.org/wiki/Logarithm Learn more about logarithms ~~in English~~ Wikipedia ]	+	[http://en.wikipedia.org/wiki/Logarithm Learn more about logarithms on Wikipedia ]

	[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/e.html Learn more about the number ''e'' in The MacTutor History of Mathematics archive ]		[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/e.html Learn more about the number ''e'' in The MacTutor History of Mathematics archive ]

Jdinglis at 20:38, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:38:14 GMT

←Older revision		Revision as of 20:38, 28 October 2008
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	{{Displayed math\|\|<math>10^{\lg 5} = 5\,\mbox{.}</math>}}		{{Displayed math\|\|<math>10^{\lg 5} = 5\,\mbox{.}</math>}}

-	~~Take~~ the natural logarithm ( ln ) of both sides.	+	Taking the natural logarithm ( <math>\ln</math>) of both sides yields
	{{Displayed math\|\|<math>\ln 10^{\lg 5} = \ln 5\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\ln 10^{\lg 5} = \ln 5\,\mbox{.}</math>}}

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	{{Displayed math\|\|<math>\lg 5 \cdot \ln 10 = \ln 5\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\lg 5 \cdot \ln 10 = \ln 5\,\mbox{.}</math>}}

-	Now divide both sides by <math>\ln 10</math> ~~giving~~ the answer	+	Now divide both sides by <math>\ln 10</math> to get the answer
	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	\lg 5 = \frac{\ln 5}{\ln 10}		\lg 5 = \frac{\ln 5}{\ln 10}

Jdinglis at 20:36, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:36:20 GMT

←Older revision		Revision as of 20:36, 28 October 2008
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	<br>		<br>
	<br>		<br>
-	By definition, ~~the~~ <math>\lg 5</math> is a number that satisfies the equality	+	By definition, <math>\lg 5</math> is a number that satisfies the equality
	{{Displayed math\|\|<math>10^{\lg 5} = 5\,\mbox{.}</math>}}		{{Displayed math\|\|<math>10^{\lg 5} = 5\,\mbox{.}</math>}}

Jdinglis at 20:35, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:35:01 GMT

←Older revision		Revision as of 20:35, 28 October 2008
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	{{Displayed math\|\|<math>		{{Displayed math\|\|<math>
	a\cdot b = 10^{\textstyle\log a} \cdot 10^{\textstyle\log b}		a\cdot b = 10^{\textstyle\log a} \cdot 10^{\textstyle\log b}
-	= \left\{ \mbox{laws of exponents} \right\}
	= 10^{\,\bbox[#AAEEFF,1pt]{\,\log a+\log b\,}}</math>}}		= 10^{\,\bbox[#AAEEFF,1pt]{\,\log a+\log b\,}}</math>}}

Jdinglis at 20:33, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:33:09 GMT

←Older revision		Revision as of 20:33, 28 October 2008
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	==The natural logarithms ==		==The natural logarithms ==

-	In practice there are two bases that are commonly used for logarithms, 10 and the number <math>e</math> <math>({}\approx 2\textrm{.}71828 \ldots\,)</math>. Logarithms using the base ''e'' are called '' natural logarithms'' and we use the notation <math>ln</math> instead of <math>\log_{\,e}</math>.	+	In practice there are two bases that are commonly used for logarithms, 10 and the number <math>e</math> <math>({}\approx 2\textrm{.}71828 \ldots\,)</math>. Logarithms using the base ''e'' are called '' natural logarithms'' and we use the notation <math>\ln</math> instead of <math>\log_{\,e}</math>.

	<div class="exempel">		<div class="exempel">

Jdinglis at 20:32, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:32:28 GMT

←Older revision		Revision as of 20:32, 28 October 2008
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	</div>		</div>

-	If the base 10 is used, one rarely writes <math>\log_{\,10}</math>, but as we have previously seen we use the notation <math> lg </math>, or simply <math>log</math>. These symbols appear on many calculators. We should also note that it makes no sense to define logarithms with base <math>1</math>.	+	If the base 10 is used, one rarely writes <math>\log_{\,10}</math>, but as we have previously seen we use the notation <math> \lg </math>, or simply <math>\log</math>. These symbols appear on many calculators. We should also note that it makes no sense to define logarithms with base <math>1</math>.

Jdinglis at 20:30, 28 October 2008

Jdinglis — Tue, 28 Oct 2008 20:30:10 GMT

←Older revision		Revision as of 20:30, 28 October 2008
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	:::"the exponent of 0.01 is -2".		:::"the exponent of 0.01 is -2".

-	This is how logarithms are defined. We ~~formalises~~ this as follows:	+	This is how logarithms are defined. We formalise this as follows:

	:::"''The logarithm'' of 1000 is 3",		:::"''The logarithm'' of 1000 is 3",