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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


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

find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
Practice Key Terms 1

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