# 4.6 Exponential and logarithmic equations  (Page 6/8)

 Page 6 / 8

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

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 and and any positive real number where if and only if Definition of a logarithm For any algebraic expression S and positive real numbers and where if and only if One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number where if and only if

## 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 $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(S\right)=c,\text{}$ 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 $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(S\right)=c.\text{\hspace{0.17em}}$ We graph both equations $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(S\right)\text{\hspace{0.17em}}$ 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] .

## 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

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this