document.write( "Question 681961: How do I solve the equation? It says to state my exact answer. If the exact answer is irrational, also give an approximate answer to three decimal places. \r
\n" ); document.write( "\n" ); document.write( "7e^(3x-2)+2=15 \n" ); document.write( "

Algebra.Com's Answer #422887 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"7e%5E%283x-2%29%2B2=15\"$
\n" ); document.write( "First isolate the base and its exponent. Subtracting 2 we get:
\n" ); document.write( " $\"7e%5E%283x-2%29=13\"$
\n" ); document.write( "Dividing by 7:
\n" ); document.write( " $\"e%5E%283x-2%29=13%2F7\"$

\n" ); document.write( "Next we will use logarithms. We will get the simplest possible expression for an answer if we match the base of the logarithm to the base of the exponent. So we will base e logarithms (i.e. ln):
\n" ); document.write( " $\"ln%28e%5E%283x-2%29%29=ln%2813%2F7%29\"$
\n" ); document.write( "Now we use a property of logarithms, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$ , which allows us to move the exponent of the argument out in front of the logarithm. (This property is the very reason we use logarithms. It allows us to move the exponent, where the variable is, out to where we can \"get at it\" with algebra.)
\n" ); document.write( " $\"%283x-2%29%2Aln%28e%29=ln%2813%2F7%29\"$
\n" ); document.write( "ln(e) = 1 by definition so this simplifies. (This is why matching the bases gets us the simplest expression.)
\n" ); document.write( " $\"3x-2=ln%2813%2F7%29\"$

\n" ); document.write( "Now we can solve for x. Adding 2:
\n" ); document.write( " $\"3x=ln%2813%2F7%29%2B2\"$
\n" ); document.write( "Dividing by 3:
\n" ); document.write( " $\"x=%28ln%2813%2F7%29%2B2%29%2F3\"$
\n" ); document.write( "This is an exact expression for the solution to your equation. (I'll leave it up to you and your calculator to find the decimal approximation.)\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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