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How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?

t = 703 , 800 , 000 × ln ( 0.8 ) ln ( 0.5 )  years    226 , 572 , 993  years .

Access these online resources for additional instruction and practice with exponential and logarithmic equations.

Key equations

One-to-one property for exponential functions For any algebraic expressions   S   and   T   and any positive real number   b ,   where
b S = b T   if and only if   S = T .
Definition of a logarithm For any algebraic expression S and positive real numbers   b     and   c ,   where   b 1 ,
log b ( S ) = c   if and only if   b c = S .
One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number   b ,   where   b 1 ,
log b S = log b T   if and only if   S = T .

Key concepts

  • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
  • When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See [link] .
  • When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See [link] , [link] , and [link] .
  • When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See [link] .
  • We can solve exponential equations with base e , by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See [link] and [link] .
  • After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See [link] .
  • When given an equation of the form log b ( S ) = c , where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation b c = S , and solve for the unknown. See [link] and [link] .
  • We can also use graphing to solve equations with the form log b ( S ) = c . We graph both equations y = log b ( S ) and y = c on the same coordinate plane and identify the solution as the x- value of the intersecting point. See [link] .
  • When given an equation of the form log b S = log b T , where S and T are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S = T for the unknown. See [link] .
  • Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See [link] .

Section exercises


How can an exponential equation be solved?

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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I'm interested in nanotube
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Ramkumar Reply
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what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
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Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
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