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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 .

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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

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.

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Questions & Answers

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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