document.write( "Question 239135: 10^(2-5x)=793 please solve without a calculator \n" ); document.write( "
Algebra.Com's Answer #175660 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"10%5E%282-5x%29=793\"
\n" ); document.write( "As far as I can tell, 793 is not an integral power of any integer. So the only way I know how to solve for x is to find the base 10 logarithm of each side:
\n" ); document.write( "\"log%28%2810%5E%282-5x%29%29%29=log%28%28793%29%29\"
\n" ); document.write( "Now we can use the property of logarithms, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", to move the exponent in the argument in front:
\n" ); document.write( "\"%282-5x%29log%28%2810%29%29=log%28%28793%29%29\"
\n" ); document.write( "and since the log(10) = 1 (This is why I chose base 10 logarithms):
\n" ); document.write( "\"2-5x+=+log%28%28793%29%29\"
\n" ); document.write( "Now that x is no longer in an exponent we can use basic Algebra to solve for it. Add -2 to (or subtract 2 from) each side:
\n" ); document.write( "\"-5x+=+-2+%2B+log%28%28793%29%29\"
\n" ); document.write( "Divide both sides by -5:
\n" ); document.write( "\"x+=+%28-2+%2B+log%28%28793%29%29%29%2F%28-5%29\"
\n" ); document.write( "or
\n" ); document.write( "\"x+=+%282+-+log%28%28793%29%29%29%2F%285%29\"

\n" ); document.write( "If \"without calculators\" means you are allowed to use tables of logarithms instead (Do textbooks come with tables of logarithms in the back anymore?), then we could do the following:
\n" ); document.write( "Factor out 100 from 793:
\n" ); document.write( "\"x+=+%282+-+log%28%28100%2A7.93%29%29%29%2F%285%29\"
\n" ); document.write( "Use the property of logarithms, \"log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29\", to split the 100 and 7.93:
\n" ); document.write( "\"x+=+%282+-+%28log%28%28100%29%29+%2B+log%28%287.93%29%29%29%29%2F%285%29\"
\n" ); document.write( "Since \"100+=+10%5E2\", \"log%28%28100%29%29+=+2\":
\n" ); document.write( "\"x+=+%282+-+%282+%2B+log%28%287.93%29%29%29%29%2F5\"
\n" ); document.write( "By factoring out the 100, we now have a logarithm we can find in a table:
\n" ); document.write( "\"x+=+%282+-+%282+%2B+0.8993%29%29%2F5\"
\n" ); document.write( "I'll leave this to you to simplify.

\n" ); document.write( "If you are not supposed to use a table, I do not see another way to get rid of the logarithm in
\n" ); document.write( "\"x+=+%282+-+log%28%28793%29%29%29%2F5\"
\n" ); document.write( "All we can do is manipulate the expression into possibly more desirable forms. As we saw before \"2+=+log%28%28100%29%29\" so we substitute for the 2:
\n" ); document.write( "\"x+=+%28log%28%28100%29%29+-+log%28%28793%29%29%29%2F%285%29\"
\n" ); document.write( "Now we can use the property of logarithms, \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\", to combine the logarithms in the numerator:
\n" ); document.write( "\"x+=+%28log%28%28100%2F793%29%29%29%2F%285%29\"
\n" ); document.write( "We can change the division by 5 into the multiplication by its reciprocal, 1/5:
\n" ); document.write( "\"x+=+%281%2F5%29%28log%28%28100%2F793%29%29%29\"
\n" ); document.write( "Then we can use the previously used property involving exponents, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", to move the number in front into the argument as an exponent:
\n" ); document.write( "\"x+=+log%28%28%28100%2F793%29%5E%281%2F5%29%29%29%29\"
\n" ); document.write( "Writing the fractional exponent in radical form we get:
\n" ); document.write( "\"x+=+log%28%28root%285%2C+%28100%2F793%29%29%29%29\"
\n" ); document.write( "

\n" ); document.write( "Which is a \"better\" answer?
\n" ); document.write( "\"x+=+%282+-+log%28%28793%29%29%29%2F%285%29\"
\n" ); document.write( "or
\n" ); document.write( "\"x+=+log%28%28root%285%2C+%28100%2F793%29%29%29%29\"
\n" ); document.write( "I can't say for sure. I prefer the first one. \n" ); document.write( "
\n" );