document.write( "Question 940336: How do we know that the equation e^x =0 has no solution? \n" ); document.write( "

Algebra.Com's Answer #573070 by MathLover1(20850) $\"\"$ $\"About$

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$\"e%5Ex=0\"$ \r
\n" ); document.write( "\n" ); document.write( "For the next step, the only option is to do the inverse function of $\"e%5Ex\"$ which is $\"natural\"$ log:\r
\n" ); document.write( "\n" ); document.write( " $\"ln%28e%5Ex%29=ln%280%29\"$
\n" ); document.write( "As you should know, we can kick the $\"x\"$ out to the front of the natural log:\r
\n" ); document.write( "\n" ); document.write( " $\"x%2Aln%28e%29=ln%280%29\"$ \r
\n" ); document.write( "\n" ); document.write( "The natural log of $\"e\"$ is of course $\"1\"$ leaving us with:
\n" ); document.write( " $\"x=ln%280%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Now, this is where we run into trouble because, as you will notice if you plug this into your calculator, you will receive an error message. This is because the natural log of $\"0\"$ can be re-written as $\"e%5Ea=0\"$ where we are solving for $\"a\"$ . However, we know $\"e\"$ represents a $\"constant\"$ ( $\"e=+2.71828\"$ ) and any constant raised to the $\"a\"$ power on the domain ( $\"-infinity\"$ , $\"infinity\"$ ) will be $\"greater\"$ than $\"0\"$ .
\n" ); document.write( "There is $\"no\"$ value for $\"a\"$ which allows the graph to cross $\"x-axis\"$ . \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-15%2C+15%2C+-15%2C+15%2C0%2C+e%5Ex%2Ce%5Ex%29+\"$ \r
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