document.write( "Question 156298: Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x. \r
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\n" ); document.write( "\n" ); document.write( "5. f(x) = ln x\r
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Algebra.Com's Answer #115099 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

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\n" ); document.write( "\n" ); document.write( "Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters \"L\" \"N\")\r
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\n" ); document.write( "\n" ); document.write( "This is the natural log of x. So it is a logarithmic function.\r
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\n" ); document.write( "\n" ); document.write( "Let's find the y value when $\"x=1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%281%29\"$ Plug in $\"x=1\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=0\"$ Take the natural log of 1 to get 0\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=1\"$ , $\"y=0\"$ . \r
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\n" ); document.write( "\n" ); document.write( "Let's find the y value when $\"x=2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%282%29\"$ Plug in $\"x=2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=0.693\"$ Take the natural log of 2 to get 0.693\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=2\"$ , $\"y=0.693\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Let's find the y value when $\"x=4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%284%29\"$ Plug in $\"x=4\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=1.386\"$ Take the natural log of 4 to get 1.386\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=4\"$ , $\"y=1.386\"$ . \r
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\n" ); document.write( "\n" ); document.write( "Let's find the y value when $\"x=8\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%288%29\"$ Plug in $\"x=8\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=2.079\"$ Take the natural log of 8 to get 2.079\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=8\"$ , $\"y=2.079\"$ . \r
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\n" ); document.write( "\n" ); document.write( "Let's find the y value when $\"x=16\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%2816%29\"$ Plug in $\"x=16\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=2.773\"$ Take the natural log of 16 to get 2.773\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=16\"$ , $\"y=2.773\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's make a table of the values we just found.\r
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Table of Values:

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document.write( "x y
1 0
2 0.693
4 1.386
8 2.079
16 2.773\r\n" );
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\n" ); document.write( "\n" ); document.write( "Since the natural log function is logarithmic, this means that the growth is logarithmic. This growth rate is slower than linear growth rate and is the slowest growth rate than all of the growth rates.\r
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\n" ); document.write( "\n" ); document.write( "So here's the order of function's growth from smallest growth rate to largest growth rate\r
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\n" ); document.write( "\n" ); document.write( " $\"y=ln%28x%29\"$ , $\"y=5x-3\"$ , $\"y=x%5E2-3x%2B2\"$ , and $\"y=2x%5E3%2B7x%5E2-x-1\"$ , and $\"y=10%5Ex\"$ \n" ); document.write( "

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