# 4.6 Exponential and logarithmic equations  (Page 2/8)

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## Rewriting equations so all powers have the same base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $\text{\hspace{0.17em}}256={4}^{x-5}.\text{\hspace{0.17em}}$ We can rewrite both sides of this equation as a power of $\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ Then we apply the rules of exponents, along with the one-to-one property, to solve for $\text{\hspace{0.17em}}x:$

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

1. Rewrite each side in the equation as a power with a common base.
2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $\text{\hspace{0.17em}}{b}^{S}={b}^{T}.$
3. Use the one-to-one property to set the exponents equal.
4. Solve the resulting equation, $\text{\hspace{0.17em}}S=T,$ for the unknown.

## Solving equations by rewriting them to have a common base

Solve $\text{\hspace{0.17em}}{8}^{x+2}={16}^{x+1}.$

Solve $\text{\hspace{0.17em}}{5}^{2x}={25}^{3x+2}.$

$x=-1$

## Solving equations by rewriting roots with fractional exponents to have a common base

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

Solve $\text{\hspace{0.17em}}{5}^{x}=\sqrt{5}.$

$x=\frac{1}{2}$

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

## Solving an equation with positive and negative powers

Solve $\text{\hspace{0.17em}}{3}^{x+1}=-2.$

This equation has no solution. There is no real value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that will make the equation a true statement because any power of a positive number is positive.

Solve $\text{\hspace{0.17em}}{2}^{x}=-100.$

The equation has no solution.

## Solving exponential equations using logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since $\text{\hspace{0.17em}}\mathrm{log}\left(a\right)=\mathrm{log}\left(b\right)\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}a=b,$ we may apply logarithms with the same base on both sides of an exponential equation.

Given an exponential equation in which a common base cannot be found, solve for the unknown.

1. Apply the logarithm of both sides of the equation.
• If one of the terms in the equation has base 10, use the common logarithm.
• If none of the terms in the equation has base 10, use the natural logarithm.
2. Use the rules of logarithms to solve for the unknown.

## Solving an equation containing powers of different bases

Solve $\text{\hspace{0.17em}}{5}^{x+2}={4}^{x}.$

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