# 4.6 Exponential and logarithmic equations

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In this section, you will:
• Use like bases to solve exponential equations.
• Use logarithms to solve exponential equations.
• Use the definition of a logarithm to solve logarithmic equations.
• Use the one-to-one property of logarithms to solve logarithmic equations.
• Solve applied problems involving exponential and logarithmic equations.

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

## Using like bases to solve exponential equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers $\text{\hspace{0.17em}}b,$ $S,$ and $\text{\hspace{0.17em}}T,$ where ${b}^{S}={b}^{T}\text{\hspace{0.17em}}$ if and only if $\text{\hspace{0.17em}}S=T.$

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, 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.

For example, consider the equation $\text{\hspace{0.17em}}{3}^{4x-7}=\frac{{3}^{2x}}{3}.\text{\hspace{0.17em}}$ To solve for $\text{\hspace{0.17em}}x,$ we use the division property of exponents to rewrite the right side so that both sides have the common base, $\text{\hspace{0.17em}}3.\text{\hspace{0.17em}}$ Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for $\text{\hspace{0.17em}}x:$

## Using the one-to-one property of exponential functions to solve exponential equations

For any algebraic expressions and any positive real number $\text{\hspace{0.17em}}b\ne 1,$

Given an exponential equation with the form $\text{\hspace{0.17em}}{b}^{S}={b}^{T},$ where $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ are algebraic expressions with an unknown, solve for the unknown.

1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $\text{\hspace{0.17em}}{b}^{S}={b}^{T}.$
2. Use the one-to-one property to set the exponents equal.
3. Solve the resulting equation, $\text{\hspace{0.17em}}S=T,$ for the unknown.

## Solving an exponential equation with a common base

Solve $\text{\hspace{0.17em}}{2}^{x-1}={2}^{2x-4}.$

Solve $\text{\hspace{0.17em}}{5}^{2x}={5}^{3x+2}.$

$x=-2$

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
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich