Logarithmic and Exponential Equations - Revision history

Khanh at 23:16, 28 October 2021

2021-10-28T23:16:31Z

Lila at 15:52, 10 October 2021

2021-10-10T15:52:41Z

Lila at 15:49, 10 October 2021

2021-10-10T15:49:41Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:36:14Z

fixing link to go straight to pdf

Rylee.taylor: added Logarithmic and Exponential Equations notes by professor Esparza

2020-08-11T22:06:26Z

added Logarithmic and Exponential Equations notes by professor Esparza

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/ Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 23:16, 28 October 2021
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	==Resources==		==Resources==
−	* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Logarithmic_and_Exponential_Functions Logarithmic and Exponential Functions], Wikibooks: A-level Mathematics
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Logarithmic_and_Exponential_Functions Logarithmic and Exponential Functions, Wikibooks: A-level Mathematics] under a CC BY-SA license

@@ Line 44: / Line 44: @@
 A '''logarithmic equation''' is an equation wherein one or more of the terms is a logarithm.
-e.g. Solve <math>\lg x + \lg (x+2) = 2</math> <ref group="note"><math>\lg</math> is another way of writing <math>\log_{10}</math></ref>
+e.g. Solve <math>\lg x + \lg (x+2) = 2</math>
 <math>\begin{align}
@@ Line 99: / Line 99: @@
 A '''power relationship''' is of the form <math>y = ax^b</math>. If we take the natural logarithm of both sides, we get <math>\ln y = \ln a + b \ln x</math>. This is a linear relationship between <math>\ln y</math> and <math>\ln x</math>.
-e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data<ref>Retrieved from [https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html NASA's Planetary Fact Sheet]</ref> to deduce this power law:
+e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data to deduce this power law:
 {| class="wikitable"

← Older revision		Revision as of 15:49, 10 October 2021
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		+	== Logarithms and Exponents ==
		+
		+	A '''logarithm''' is the inverse function of an exponent.
		+
		+	e.g. The inverse of the function <math>f(x) = 3^x</math> is <math>f^{-1}(x) = \log_3 x</math>.
		+
		+	In general, <math>y = b^x \iff x = \log_b y</math>, given that <math>b > 0</math>.
		+
		+	== Laws of Logarithms ==
		+
		+	The laws of logarithms can be derived from the laws of exponentiation:
		+
		+	<math>\begin{align}
		+	x^{a+b} = x^a \times x^b &\iff \log a + \log b = \log ab \\
		+	x^{a-b} = x^a \div x^b &\iff \log a - \log b = \log a/b \\
		+	(x^a)^b = x^{ab} &\iff \log a^b = b \log a
		+	\end{align}</math>
		+
		+	These laws apply to logarithms of any given base
		+
		+	== Natural Logarithms ==
		+
		+	The '''natural logarithm''' is a logarithm with base <math>e</math>, where <math>e</math> is a constant such that the function <math>e^x</math> is its own derivative.
		+
		+	The natural logarithm has a special symbol: <math>\ln x</math>
		+
		+	The graph <math>y = e^{kx}</math> exhibits exponential growth when <math>k > 0</math> and exponential decay when <math>k < 0</math>. The inverse graph is <math>y = \frac{1}{k} \ln x</math>. [https://www.desmos.com/calculator/azbv4kz38z Here] is an interactive graph which shows the two functions as inverses of one another.
		+
		+	== Solving Logarithmic and Exponential Equations ==
		+
		+	An '''exponential equation''' is an equation in which one or more of the terms is an exponential function. e.g. <math>5^x = 2^{x+2}</math>. Exponential equations can be solved with logarithms.
		+
		+	e.g. Solve <math>3^{x+1} = 4^{2x-1}</math>
		+
		+	<math>\begin{align}
		+	3^{x+1} &= 4^{2x-1} \\
		+	(x+1)\ln 3 &= (2x-1)\ln 4 \\
		+	x\ln 3 + \ln 3 &= 2x\ln 4 - \ln 4 \\
		+	\ln 3 + \ln 4 &= x(2\ln 4 - \ln 3) \\
		+	x &= \frac{\ln 3 + \ln 4}{2\ln 4 - \ln 3} \\
		+	x &\approx 1.4844
		+	\end{align}</math>
		+
		+	A '''logarithmic equation''' is an equation wherein one or more of the terms is a logarithm.
		+
		+	e.g. Solve <math>\lg x + \lg (x+2) = 2</math> <ref group="note"><math>\lg</math> is another way of writing <math>\log_{10}</math></ref>
		+
		+	<math>\begin{align}
		+	\lg x + \lg (x+2) &= 2 \\
		+	\lg (x(x+2)) &= 2 \\
		+	x(x+2) &= 100 \\
		+	x^2 + 2x &= 100 \\
		+	(x + 1)^2 &= 101 \\
		+	x+1 &= \sqrt{101} \\
		+	x &= -1 \pm \sqrt{101}
		+	\end{align}</math>
		+
		+	== Converting Relationships to a Linear Form ==
		+
		+	In maths and science, it is easier to deal with linear relationships than non-linear relationships. Logarithms can be used to convert some non-linear relationships into linear relationships.
		+
		+	=== Exponential Relationships ===
		+
		+	An '''exponential relationship''' is of the form <math>y = ab^x</math>. If we take the natural logarithm of both sides, we get <math>\ln y = \ln a + x \ln b</math>. We now have a linear relationship between <math>\ln y</math> and <math>x</math>.
		+
		+	e.g. The following data is related with an exponential relationship. Determine this exponential relationship, then convert it to linear form.
		+
		+	{\| style="width:30%" class="wikitable"
		+	\|-
		+	! x !! y
		+	\|-
		+	\| 0 \|\| 5
		+	\|-
		+	\| 2 \|\| 45
		+	\|-
		+	\| 4 \|\| 405
		+	\|}
		+
		+	<math>\begin{align}
		+	\text{Exponential relationship } \implies y &= ab^x \\
		+	5 &= ab^0 = a(1) \\
		+	a &= 5 \\
		+	y &= 5b^x \\
		+	45 &= 5b^2 \\
		+	9 &= b^2 \\
		+	b &= 3 \\
		+	y &= 5(3^x)
		+	\end{align}</math>
		+
		+	Now convert it to linear form by taking the natural logarithm of both sides:
		+
		+	<math>\begin{align}
		+	y &= 5(3^x) \\
		+	\ln y &= \ln 5 + x \ln 3
		+	\end{align}</math>
		+
		+	=== Power Relationships ===
		+
		+	A '''power relationship''' is of the form <math>y = ax^b</math>. If we take the natural logarithm of both sides, we get <math>\ln y = \ln a + b \ln x</math>. This is a linear relationship between <math>\ln y</math> and <math>\ln x</math>.
		+
		+	e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data<ref>Retrieved from [https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html NASA's Planetary Fact Sheet]</ref> to deduce this power law:
		+
		+	{\| class="wikitable"
		+	\|-
		+	! Planet !! Distance from Sun /10<sup>6</sup> km !! Orbital Period /days
		+	\|-
		+	\| Earth \|\| 149.6 \|\| 365.2
		+	\|-
		+	\| Mars \|\| 227.9 \|\| 687.0
		+	\|-
		+	\| Jupiter \|\| 778.6 \|\| 4331
		+	\|}
		+
		+	<math>\begin{align}
		+	\text{Power law}\implies T &= aR^b \\
		+	\text{Use Earth data}\implies 365.2 &= a(149.6^b) \\
		+	\ln 365.2 &= \ln a + b \ln 149.6 \\
		+	\text{Use Mars data}\implies 687.0 &= a(227.9^b) \\
		+	\ln 687.0 &= \ln a + b \ln 227.9 \\
		+	\ln 687.0 - \ln 365.2 &= \ln a - \ln a + b\ln 227.9 - b\ln 149.6 \\
		+	\ln \frac{687.0}{365.2} &= 0 + b(\ln\frac{227.9}{149.6}) \\
		+	b &= \frac{\ln \tfrac{687.0}{365.2}}{\ln\tfrac{227.9}{149.6}} \approx 1.5011 \\
		+	\ln 365.2 &= \ln a + 1.5011 \ln 149.6 \\
		+	\ln a &= \ln 365.2 - \ln 1839.9 \\
		+	\ln a &= \ln 0.1985 \\
		+	a &= 0.1985 \\
		+	\therefore T &= 0.1985R^{1.5011}
		+	\end{align}</math>
		+
		+	==Resources==
		+	* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Logarithmic_and_Exponential_Functions Logarithmic and Exponential Functions], Wikibooks: A-level Mathematics
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 22:36, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/ Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA.