document.write( "Question 616355: Please help me with this word problem: The probability of a transistor failing within t months is given by P(t)=1-e^-0.016t. use logarithms to determine within how many months the probability of failure will be 99.9% Round 2 decimal places. \n" ); document.write( "

Algebra.Com's Answer #387657 by jsmallt9(3758) $\"\"$ $\"About$

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$\"P%28t%29+=+1+-+e%5E%28-0.016t%29\"$
\n" ); document.write( "Your problem, in essence, is to find the time for a certain probability. So we will substitute the probability into the equation above and solve for t.

\n" ); document.write( "In order to enter the probability into the equation correctly we must realize that you should never do calculations with percents. Always use the decimal or fraction equivalent. So we will use 0.999 (which is the decimal for 99.9%):
\n" ); document.write( " $\"0.999+=+1+-+e%5E%28-0.016t%29\"$
\n" ); document.write( "Now we solve for t.

\n" ); document.write( "Solving for a variable that is in an exponent, like t, usually starts with isolating the base and its exponent. So we will subtract 1 from each side:
\n" ); document.write( " $\"-0.001+=+-e%5E%28-0.016t%29\"$
\n" ); document.write( "And then we'll divide both sides by -1 to get rid of the \"-\":
\n" ); document.write( " $\"0.001+=+e%5E%28-0.016t%29\"$
\n" ); document.write( "The base, e, and its exponent, -0.016t, are now isolated on the right side.

\n" ); document.write( "The next step is to use logarithms. Logarithms of any base may be used. But there are advantages to choosing certain bases:

Choosing a base for the logarithm that matches the base of the exponent will result in the simplest possible answer.
Choosing a base that your calculator \"knows\", base 10 or base e (aka ln), will result in a expression that can easily be converted to a decimal approximation.

With this equation we can choose a base that both matches the exponent's base and can easily be converted to decimals by choosing base e (aka ln) logarithms! So we will find the ln of each side:
\n" ); document.write( " $\"ln%280.001%29+=+ln%28e%5E%28-0.016t%29%29\"$

\n" ); document.write( "Next we use a property of logarithms, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$ , to \"move\" the exponent in 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 in front. With the vriable out of an exponent we can now solve for it. Using this property on our equation we get:
\n" ); document.write( " $\"ln%280.001%29+=+%28-0.016t%29ln%28e%29\"$

\n" ); document.write( "By definition $\"log%28A%2C+%28a%29%29+=+1\"$ for all bases. So ln(e) = 1. This makes our equation:
\n" ); document.write( " $\"ln%280.001%29+=+-0.016t\"$

\n" ); document.write( "All we have left is to divide by -0.016:
\n" ); document.write( " $\"ln%280.001%29%2F%28-0.016%29+=+t\"$
\n" ); document.write( "This is an exact expression for the solution to your equation. Since you were asked for a rounded decimal answer we get out our calculators:
\n" ); document.write( " $\"ln%280.001%29%2F%28-0.016%29+=+t\"$
\n" ); document.write( " $\"%28-6.90775528%29%2F%28-0.016%29+=+t\"$
\n" ); document.write( " $\"431.73470494+=+t\"$
\n" ); document.write( "Rounded to two decimal places:
\n" ); document.write( " $\"431.73+=+t\"$
\n" ); document.write( "So there will be a 99.9% chance of transistor failure within almost 432 months (which is almost 36 years). \n" ); document.write( "

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