L4.2 B1 Logarithmic and Exponential Equations
Completion requirements
Unit 4
Logarithms
Read
Part 4.2B corresponds to section 8.4, starting on page 404 of your Pre-Calculus 12 textbook.
There are a number of strategies that can be applied to solve
logarithmic equations, including rewriting a logarithmic equation in
exponential form, using technology to solve a logarithmic equation
graphically, simplifying one or both sides of the equation using the
laws of logarithms, taking the logarithm of both sides of the equation,
and using the LOG function on a calculator, to name only a few.
Regardless of the strategy used, it is imperative that you identify any restrictions on the variable, and then confirm your solution(s).
Regardless of the strategy used, it is imperative that you identify any restrictions on the variable, and then confirm your solution(s).
Algebraic Solutions
To solve algebraically means to manipulate the equation to isolate the desired variable.
Solve «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mi»x«/mi»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»4«/mn»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«/mrow»«/mstyle»«/math»
algebraically.
First, note that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» is the restriction on the variable.
Manipulate the equation so there are single logarithmic expressions on both sides of the equal sign.
Because the logarithms have the same base, and logarithmic functions are one-to-one, the arguments for each must be the same for the equation to be balanced.
So,
The value «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfies the variable restriction of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», and thus «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» is a solution to the equation.
First, note that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» is the restriction on the variable.
Manipulate the equation so there are single logarithmic expressions on both sides of the equal sign.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable columnalign=¨right center
left¨»«mtr»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mi»x«/mi»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»4«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«mi»x«/mi»«mo»§#8226;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«mn»4«/mn»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Because the logarithms have the same base, and logarithmic functions are one-to-one, the arguments for each must be the same for the equation to be balanced.
So,
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable columnalign=¨right center
left¨»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»12«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The value «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfies the variable restriction of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», and thus «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» is a solution to the equation.
The same equation could have been solved by converting the logarithmic equation to an exponential equation.
Algebraically solve the equation «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mi»x«/mi»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»4«/mn»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«/mrow»«/mstyle»«/math»
by converting to an
exponential equation.
Bring all terms to the left side of the equation.
Simplify the left side to a single logarithmic expression using logarithmic laws.
Now, write the logarithmic equation as an exponential equation. Then, solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
Note either method is correct and the solution «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfies the restriction of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
Bring all terms to the left side of the equation.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mi»x«/mi»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»4«/mn»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mn»12«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
Simplify the left side to a single logarithmic expression using logarithmic laws.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable
columnalign=¨right center
left¨»«mtr»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfrac»«mrow»«mn»4«/mn»«mi»x«/mi»«/mrow»«mn»12«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfrac»«mi»x«/mi»«mn»3«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Now, write the logarithmic equation as an exponential equation. Then, solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable
columnalign=¨right center
left¨»«mtr»«mtd»«msup»«mn»7«/mn»«mn»0«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»x«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»1«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»x«/mi»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Note either method is correct and the solution «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» satisfies the restriction of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
Solving Exponential Equations
Because logarithms and exponents are related, and exponents can be used to solve logarithmic equations, it follows that logarithms can be used to solve exponential equations.
Solve «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«msup»«mn»6«/mn»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»=«/mo»«msup»«mn»8«/mn»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/msup»«/mrow»«/mstyle»«/math».
The two sides of the equation have different bases. It is difficult to
write the two expressions using the same base, so we will use logarithms
to solve this equation.
Take the base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» logarithm of both sides of the equation.
Note: It does not matter what logarithmic base we use because as long as the same operation is done to both sides of the equation, the equation remains balanced. The common logarithm is selected because most calculators use base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
Now, use the power law of logarithms to bring the exponents of the arguments down.
Now, distribute through the brackets on both sides.
Then, collect like terms.
Factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» out of the terms on the left side.
Then, divide both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»3«/mn»«mi»log«/mi»«mn»6«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math» to get «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» by itself on the left.
A calculator can be used to determine a decimal approximation of the solution.
If rounded to two decimal places, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8784;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»35«/mn»«/mrow»«/mstyle»«/math».
Note the use of brackets in the calculator entry to ensure the numerator and denominator are kept separate and evaluated properly.
Take the base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» logarithm of both sides of the equation.
Note: It does not matter what logarithmic base we use because as long as the same operation is done to both sides of the equation, the equation remains balanced. The common logarithm is selected because most calculators use base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»log«/mi»«msup»«mn»6«/mn»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»=«/mo»«mi»log«/mi»«msup»«mn»8«/mn»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/msup»«/mrow»«/mstyle»«/math»
Now, use the power law of logarithms to bring the exponents of the arguments down.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mi»log«/mi»«mn»6«/mn»«mo»=«/mo»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math»
Now, distribute through the brackets on both sides.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mn»3«/mn»«mi»x«/mi»«mi»log«/mi»«mn»6«/mn»«mo»+«/mo»«mi»log«/mi»«mn»6«/mn»«mo»=«/mo»«mi»x«/mi»«mi»log«/mi»«mn»8«/mn»«mo»+«/mo»«mn»3«/mn»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math»
Then, collect like terms.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mn»3«/mn»«mi»x«/mi»«mi»log«/mi»«mn»6«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«mi»log«/mi»«mn»8«/mn»«mo»=«/mo»«mn»3«/mn»«mi»log«/mi»«mn»8«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»6«/mn»«/mrow»«/mstyle»«/math»
Factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» out of the terms on the left side.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mfenced»«mrow»«mn»3«/mn»«mi»log«/mi»«mn»6«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»log«/mi»«mn»8«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»6«/mn»«/mrow»«/mstyle»«/math»
Then, divide both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»3«/mn»«mi»log«/mi»«mn»6«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math» to get «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» by itself on the left.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»3«/mn»«mi»log«/mi»«mn»8«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»6«/mn»«/mrow»«mrow»«mn»3«/mn»«mi»log«/mi»«mn»6«/mn»«mo»§#8722;«/mo»«mi»log«/mi»«mn»8«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
A calculator can be used to determine a decimal approximation of the solution.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»349«/mn»«mo»§#8230;«/mo»«/mrow»«/mstyle»«/math»
If rounded to two decimal places, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8784;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»35«/mn»«/mrow»«/mstyle»«/math».
Note the use of brackets in the calculator entry to ensure the numerator and denominator are kept separate and evaluated properly.
Logarithms can be useful in solving problems that are related to exponential modelling.
A
lab technician is testing the effectiveness of an
antibiotic against a
particular strain of bacteria. Her measurements
indicate her test
culture initially had «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math»
bacteria/mL, and dropped to «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»200«/mn»«/mrow»«/mstyle»«/math»
bacteria/mL «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mn»6«/mn»«/mstyle»«/math» hours after
adding the antibiotic.
To the nearest hundredth of an hour, what is the half-life of the bacteria in the presence of the antibiotic? That is, how long does it take for the bacteria concentration to decrease by half?
To the nearest hundredth of an hour, what is the half-life of the bacteria in the presence of the antibiotic? That is, how long does it take for the bacteria concentration to decrease by half?
The half-life situation described is one of
exponential decay of the
form «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»N«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«msub»«mi»N«/mi»«mi»O«/mi»«/msub»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mi»t«/mi»«mi»h«/mi»«/mfrac»«/msup»«/mrow»«/mstyle»«/math»,
where «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«msub»«mi»N«/mi»«mi»O«/mi»«/msub»«/mstyle»«/math» is the
initial number of bacteria
per mL, «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» is the time
that has passed, in hours, «math
style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math» is the
half-life of the bacteria, in hours, and
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»N«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mrow»«/mstyle»«/math»
is the number of bacteria remaining after
time «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math».
Isolate the power by dividing both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math».
Now, take the common logarithm of both sides of the equation, and apply logarithmic laws.
Multiply both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math», and then divide by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo».«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
Then, determine the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math» by using the calculator.
The bacteria has a half life of approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»14«/mn»«/mrow»«/mstyle»«/math» hours. After the antibiotic is administered, it will take about «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»14«/mn»«/mrow»«/mstyle»«/math» hours for the bacteria concentration to drop to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math» bacteria/mL.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«msub»«mi»N«/mi»«mi»O«/mi»«/msub»«mo»=«/mo»«mn»2«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«mo»=«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»h«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»N«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»200«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mn»1«/mn»«mspace
width=¨0.33em¨/»«mn»200«/mn»«mo»=«/mo»«mn»2«/mn»«mspace
width=¨0.33em¨/»«mn»000«/mn»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/msup»«/mrow»«/mstyle»«/math»
Isolate the power by dividing both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable columnalign=¨right center
left¨»«mtr»«mtd»«mfrac»«mrow»«mn»1«/mn»«mspace
width=¨0.33em¨/»«mn»200«/mn»«/mrow»«mrow»«mn»2«/mn»«mspace
width=¨0.33em¨/»«mn»000«/mn»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/msup»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Now, take the common logarithm of both sides of the equation, and apply logarithmic laws.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable columnalign=¨right center
left¨»«mtr»«mtd»«mn»0«/mn»«mo».«/mo»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»log«/mi»«mn»0«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»log«/mi»«msup»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»log«/mi»«mn»0«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mfrac»«mn»6«/mn»«mi»h«/mi»«/mfrac»«/mfenced»«mi»log«/mi»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Multiply both sides by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math», and then divide by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo».«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mtable columnalign=¨right center
left¨»«mtr»«mtd»«mi»h«/mi»«mi»log«/mi»«mn»0«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»log«/mi»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»h«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»6«/mn»«mi»log«/mi»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mrow»«mrow»«mi»log«/mi»«mn»0«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»6«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Then, determine the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»h«/mi»«/mstyle»«/math» by using the calculator.
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo»§#8784;«/mo»«mn»8«/mn»«mi
mathvariant=¨normal¨».«/mi»«mn»14«/mn»«/mrow»«/mstyle»«/math» hours.
The bacteria has a half life of approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»14«/mn»«/mrow»«/mstyle»«/math» hours. After the antibiotic is administered, it will take about «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»14«/mn»«/mrow»«/mstyle»«/math» hours for the bacteria concentration to drop to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math» bacteria/mL.