document.write( "Question 347395: Assume that the number of viruses present in a sample is modeled by the exponential function \"f(t) = 10t,\" where t is the elapsed time in minutes. How would you apply logarithms to determine when the sample will grow to 5 billion viruses?
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Algebra.Com's Answer #248420 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
I assume the function is:
\n" ); document.write( " $\"f%28t%29+=+10%5Et\"$
\n" ); document.write( "If so, please use \"^\" (shift 6) to indicate exponents in the future.

\n" ); document.write( "To find when the sample will grow to 5 billion, we replace the function value with 5 billion and solve for t. I will use scientific notation for the 5 billion because it will make the problem a little easier:
\n" ); document.write( " $\"5+%2A+10%5E9+=+10%5Et\"$
\n" ); document.write( "Now we solve for t. To solve for a variable which is in an exponent, you usually use logarithms. The base of logarithm we will use will be base 10. Finding the base 10 logarithm of each side we get:
\n" ); document.write( " $\"log%28%285+%2A+10%5E9%29%29=+log%28%2810%5Et%29%29\"$
\n" ); document.write( "Now we can use properties of logarithms to solve for t. First I will use the property, $\"log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29\"$ , to split the log on the left into two logs:
\n" ); document.write( " $\"log%28%285%29%29+%2B+log%28%2810%5E9%29%29=+log%28%2810%5Et%29%29\"$
\n" ); document.write( "Next I will use another property of logarithms, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$ , on the two logs with exponents in their arguments. Using this property is the primary reason for using logarithms on equations like this. The property allows us the move the exponent with a variable in it out in front. In simple terms, we get to move the variable out of the exponent \"where we can get at it\".
\n" ); document.write( " $\"log%28%285%29%29+%2B+9%2Alog%28%2810%29%29=+t%2Alog%28%2810%29%29\"$
\n" ); document.write( "By definition, log(10) is 1. So the equation simplifies to:
\n" ); document.write( " $\"log%28%285%29%29+%2B+9+=+t\"$
\n" ); document.write( "This is an exact expression for the number of minutes it will take for the sample to grow to 5 billion. If you want a decimal approximation for the answer, ask your calculator to find log(5) and add that result to 9.

\n" ); document.write( "It's a pretty amazing statement of exponential functions to find that a single virus could grow to 5 billion in less than 10 minutes!\r
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