document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200811>Question 1200811</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write a demand and a supply curve expressed in the form of an exponential and<BR>\n" );
document.write( "logarithmic functions respectively. Demonstrate graphically or otherwise, that the<BR>\n" );
document.write( "functions reflect all the characteristics of a demand and supply curve. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848038 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848038\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Demand Curve (Exponential)**\r<BR>\n" );
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document.write( "* **Function:** <BR>\n" );
document.write( "   * **Q_d = a * e^(-bP)** <BR>\n" );
document.write( "      * Where:<BR>\n" );
document.write( "          * Q_d is the quantity demanded<BR>\n" );
document.write( "          * P is the price<BR>\n" );
document.write( "          * a and b are positive constants\r<BR>\n" );
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document.write( "* **Characteristics:**<BR>\n" );
document.write( "    * **Negative Slope:** As price (P) increases, the quantity demanded (Q_d) exponentially decreases. This captures the inverse relationship between price and quantity demanded.<BR>\n" );
document.write( "    * **Asymptotic to the Price Axis:** As price increases indefinitely, the quantity demanded approaches zero but never reaches it.\r<BR>\n" );
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document.write( "**2. Supply Curve (Logarithmic)**\r<BR>\n" );
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document.write( "* **Function:**<BR>\n" );
document.write( "   * **Q_s = c * ln(P) + d**<BR>\n" );
document.write( "      * Where:<BR>\n" );
document.write( "          * Q_s is the quantity supplied<BR>\n" );
document.write( "          * P is the price<BR>\n" );
document.write( "          * c and d are constants (c should be positive to ensure a positive slope)\r<BR>\n" );
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document.write( "* **Characteristics:**<BR>\n" );
document.write( "    * **Positive Slope:** As price (P) increases, the quantity supplied (Q_s) increases logarithmically. This reflects the direct relationship between price and quantity supplied.<BR>\n" );
document.write( "    * **Increasing at a Decreasing Rate:** The rate of increase in quantity supplied slows down as price increases.\r<BR>\n" );
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document.write( "**Demonstrating Graphically**\r<BR>\n" );
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document.write( "1. **Choose Values for Constants:** Select appropriate values for the constants (a, b, c, and d) in the demand and supply functions to create realistic curves. For example:<BR>\n" );
document.write( "   * **Demand:** Q_d = 100 * e^(-0.05P)<BR>\n" );
document.write( "   * **Supply:** Q_s = 20 * ln(P) + 50\r<BR>\n" );
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document.write( "2. **Plot the Curves:** <BR>\n" );
document.write( "   * Use graphing software (like Excel, Desmos, or a graphing calculator) to plot the demand and supply curves on the same graph. <BR>\n" );
document.write( "   * **X-axis:** Price (P)<BR>\n" );
document.write( "   * **Y-axis:** Quantity (Q_d or Q_s)\r<BR>\n" );
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document.write( "3. **Observe Characteristics:**<BR>\n" );
document.write( "   * **Demand Curve:** The curve should slope downward, showing a decrease in quantity demanded as price increases. It should approach but never touch the price axis.<BR>\n" );
document.write( "   * **Supply Curve:** The curve should slope upward, showing an increase in quantity supplied as price increases. The slope should gradually decrease as price increases.\r<BR>\n" );
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document.write( "4. **Equilibrium:**<BR>\n" );
document.write( "   * Identify the point where the demand and supply curves intersect. This point represents the equilibrium price and quantity.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* These are simplified examples. Real-world demand and supply curves can be more complex and may involve other factors like consumer preferences, input costs, technology, and government regulations.<BR>\n" );
document.write( "* The specific values of the constants (a, b, c, and d) will significantly impact the shape and position of the curves.\r<BR>\n" );
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document.write( "By following these steps, you can effectively demonstrate the characteristics of demand and supply curves using exponential and logarithmic functions.<BR>\n" );
document.write( " </SPAN>\n" );
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