document.write( "Question 718243: What is the approximate solution to 5^6x+3=37? This means 5 with the exponents of 5 being 6x+3 and the answer equaling 37.
\n" ); document.write( "Any help is appreciated.\r
\n" ); document.write( "\n" ); document.write( "Thanks,\r
\n" ); document.write( "\n" ); document.write( "Sherry \n" ); document.write( "
Algebra.Com's Answer #440851 by DrBeeee(684)\"\" \"About 
You can put this solution on YOUR website!
When your variable is in an exponent, the general rule is that you use logarithms to solve for the variable. For example, if we have a power (remember your definition of a power?) equal to a number as
\n" ); document.write( "(1) 2^x = 3
\n" ); document.write( "The procedure is to take the LN of both sides of the equation and get
\n" ); document.write( "(2) LN(2^x) = LN(3)
\n" ); document.write( "Now we apply the rule that the LN of a power is equal to the exponent times the LN of the base and get
\n" ); document.write( "(3) x*LN(2) = LN(3)
\n" ); document.write( "Now you treat the LN(2) and LN(3) just as you would we any real number and get
\n" ); document.write( "(4) x = LN(3)/LN(2)
\n" ); document.write( "Now use your scienfific calculator to get
\n" ); document.write( "(5) x = 1.0986.../0.6931... or
\n" ); document.write( "(6) x = 1.584...
\n" ); document.write( "Rarely will it be a \"nice\" round number, so keep all intermediate computational values in the calculator; rounding only the final answer.
\n" ); document.write( "So much for the lesson, now let's do your problem
\n" ); document.write( "(7) \"5%5E%286x%2B3%29+=+37\"
\n" ); document.write( "Note we have a power on the left and a real number on the right just as we do in example (1).
\n" ); document.write( "Now take the LN of (7) and get
\n" ); document.write( "(8) \"%286x%2B3%29%2ALN%285%29+=+LN%2837%29\" or
\n" ); document.write( "(9) \"%286x%2B3%29+=+LN%2837%29%2FLN%285%29\" or
\n" ); document.write( "(10) \"6x+=+%28LN%2837%29%2FLN%285%29%29-3\" or
\n" ); document.write( "(11) \"x+=+%28%28LN%2837%29%2FLN%285%29%29-3%29%2F6\" or
\n" ); document.write( "(12) x = (((3.6109...)/(1.6094...))-3)/6 or
\n" ); document.write( "(13) x = ((2.243...)-3)/6 or
\n" ); document.write( "(14) x = (-0.7564...)/6 or
\n" ); document.write( "(15) x = -0.126...
\n" ); document.write( "Let's check this using (7).
\n" ); document.write( "Is (5^(6*(-0.126...)+3) = 37)?
\n" ); document.write( "Is (5^(2.243...) = 37)?
\n" ); document.write( "Is (37 = 37)? Yes
\n" ); document.write( "Answer: x is approximately -0.1260684
\n" ); document.write( "PS In the above calculations you could use LOG instead of LN, it doen't matter. \n" ); document.write( "
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