For any algebraic expressions
$\text{}S\text{}$ and
$\text{}T\text{}$ and any positive real number
$\text{}b,\text{}$ where
${b}^{S}={b}^{T}\text{}$ if and only if
$\text{}S=T.$
Definition of a logarithm
For any algebraic expression
S and positive real numbers
$\text{}b\text{}$ and
$\text{}c,\text{}$ where
$\text{}b\ne 1,$ ${\mathrm{log}}_{b}(S)=c\text{}$ if and only if
$\text{}{b}^{c}=S.$
One-to-one property for logarithmic functions
For any algebraic expressions
S and
T and any positive real number
$\text{}b,\text{}$ where
$\text{}b\ne 1,$ ${\mathrm{log}}_{b}S={\mathrm{log}}_{b}T\text{}$ if and only if
$\text{}S=T.$
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
$\text{\hspace{0.17em}}e,$ 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
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}(S)=c,\text{}$ where
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
$\text{\hspace{0.17em}}{b}^{c}=S,\text{}$ and solve for the unknown. See
[link] and
[link] .
We can also use graphing to solve equations with the form
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}(S)=c.\text{\hspace{0.17em}}$ We graph both equations
$\text{\hspace{0.17em}}y={\mathrm{log}}_{b}(S)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y=c\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T,\text{}$ where
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
$\text{\hspace{0.17em}}S=T\text{\hspace{0.17em}}$ 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
Verbal
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?
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.
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.