what common base can be used to rewrite each side of the equation mc002-1.jpg 2 3 5 8
Learning Objectives
- Exponential Equations with Unlike Bases
- Identify an exponential equation whose terms all have the same base
- Idenitfy cases where equations tin be rewritten then all terms take the same base
- Apply the i-to-one holding of exponents to solve an exponential equation
- Exponential Equations with Unlike Bases
- Apply logarithms to solve exponential equations whose terms cannot be rewritten with the same base
- Solve exponential equations of the form [latex]y=A{e}^{kt}[/latex] for t
- Recognize when there may be extraneous solutions, or no solutions for exponential equations
- Logarithmic Equations
- Use the definition of a logarithm to solve logarithmic equations
- Employ a graph to verify or analyze the solution to a logarithmic equation
- Applied Exponential and Logarithmic Equations
- Solve one-half-life problems
- Solve pH issues
- Solve problems involving Richter scale readings
- Solve problems involving decibels
Wild rabbits in Commonwealth of australia. The rabbit population grew and so speedily in Australia that the upshot became known as the "rabbit plague." (credit: Richard Taylor, Flickr)
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 tin be solved to analyze and make predictions about exponential growth. In this department, we will acquire techniques for solving exponential functions.
When an exponential equation has the aforementioned base on each side, the exponents must be equal. This too applies when the exponents are algebraic expressions. Therefore, nosotros tin can solve many exponential equations past using the rules of exponents to rewrite each side every bit a power with the same base. Then, we can set the exponents equal to one another, and solve for the unknown.
For case, consider the equation [latex]{3}^{4x - 7}=\frac{{3}^{2x}}{3}[/latex]. To solve for x, we utilise the division property of exponents to rewrite the correct side so that both sides have the common base of operations, three. So we apply the one-to-one property of exponents past setting the exponents equal to one some other and solving for x:
[latex]\brainstorm{array}{c}{3}^{4x - 7}\hfill & =\frac{{3}^{2x}}{3}\hfill & \hfill \\ {3}^{4x - 7}\hfill & =\frac{{3}^{2x}}{{3}^{i}}\hfill & {\text{Rewrite iii as 3}}^{1}.\hfill \\ {3}^{4x - 7}\hfill & ={3}^{2x - 1}\hfill & \text{Use the division holding of exponents}\text{.}\hfill \\ 4x - 7\hfill & =2x - one\text{ }\hfill & \text{Apply the i-to-one property of exponents}\text{.}\hfill \\ 2x\hfill & =half dozen\hfill & \text{Subtract 2}x\text{ and add 7 to both sides}\text{.}\hfill \\ x\hfill & =3\hfill & \text{Divide by 3}\text{.}\hfill \end{array}[/latex]
In our first instance, nosotros solve an exponential equation whose terms all have a mutual base.
Case
Solve [latex]{2}^{10 - 1}={2}^{2x - 4}[/latex].
In general, we tin summarize solving exponential equations whose terms all have the aforementioned base of operations in this fashion:
For any algebraic expressions S and T, and any positive real number [latex]b\ne 1[/latex]
[latex]{b}^{S}={b}^{T}\text{ if and only if }S=T[/latex]
- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the course [latex]{b}^{S}={b}^{T}[/latex].
- Use the one-to-one property to set the exponents equal.
- Solve the resulting equation, Southward= T, for the unknown.
Rewriting Equations And then All Powers Have the Aforementioned Base
Sometimes we can rewrite the terms in an equation as powers with a mutual base of operations, and solve using the one-to-1 property. This takes a keen centre for recognizing mutual powers. For instance, y'all tin rewrite eight as [latex]2^three[/latex] or 36 as [latex]6^two[/latex] or [latex]\frac{i}{4}[/latex] as [latex]\left(\frac{1}{2}\right)^{2}[/latex]
Consider the equation [latex]256={4}^{10 - five}[/latex]. We tin rewrite both sides of this equation every bit a power of two. Then we apply the rules of exponents, forth with the i-to-one property, to solve for ten:
[latex]\begin{array}{c}256={four}^{10 - five}\hfill & \hfill \\ {2}^{eight}={\left({two}^{2}\right)}^{10 - five}\hfill & \text{Rewrite each side equally a ability with base 2}.\hfill \\ {2}^{viii}={two}^{2x - 10}\hfill & \text{Use the one-to-one belongings of exponents}.\hfill \\ eight=2x - 10\hfill & \text{Apply the ane-to-ane property of exponents}.\hfill \\ eighteen=2x\hfill & \text{Add 10 to both sides}.\hfill \\ x=9\hfill & \text{Dissever by ii}.\hfill \finish{array}[/latex]
In the adjacent example, we show how to observe a mutual base of operations for ii expressions whose bases are 8, and 16. Nosotros tin so solve the resulting equation using the one-to-one property of exponents.
Example
Solve [latex]{8}^{x+ii}={16}^{x+1}[/latex].
In our adjacent instance, we are given an exponential equation that contains a square root. Recollect that you tin can write roots as rational exponents, so you may be able to find like bases when it is not completely obvious at offset.
Example
Solve [latex]{2}^{5x}=\sqrt{2}[/latex].
By irresolute [latex]\sqrt{2}[/latex] to [latex]{2}^{\frac{i}{2}}[/latex] we were able to solve the equation in the previous example. In general, here are some steps to consider when you lot are solving exponential equations. A good first footstep is always to decide whether y'all can rewrite the terms with a mutual base.
- Rewrite each side in the equation as a ability with a common base.
- Apply the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{Due south}={b}^{T}[/latex].
- Use the ane-to-one belongings to prepare the exponents equal.
- Solve the resulting equation, Southward= T, for the unknown.
Recall Almost It
Do all exponential equations have a solution? If non, how can we tell if at that place is a solution during the problem-solving process? Write your thoughts in the textbox below before you check our proposed answer.
Show Answer
No. Recall that the range of an exponential office is always positive. While solving the equation, we may obtain an expression that is undefined.
In the side by side case nosotros testify you a case where there is no solution to an exponential equation. Retrieve how exponential functions are defined and ask yourself – "does this make sense" before diving into solving exponential equations.
Example
Solve [latex]{iii}^{ten+1}=-2[/latex].
Analysis of the Solution
Exponential Equations with unlike Bases
Sometimes the terms of an exponential equation cannot be rewritten with a mutual base of operations. In these cases, nosotros solve by taking the logarithm of each side. Retrieve, since [latex]\mathrm{log}\left(a\right)=\mathrm{log}\left(b\correct)[/latex] is equivalent to a= b, we may use logarithms with the same base on both sides of an exponential equation.
In our first example we will use the law of logs combined with factoring to solve an exponential equation whose terms do not have the same base. Notation how first, we rewrite the exponential terms as logarithms.
Case
Solve [latex]{5}^{ten+2}={4}^{x}[/latex].
In full general we tin can solve exponential equations whose terms do not accept like bases in the following way:
- Apply the logarithm of both sides of the equation.
- If 1 of the terms in the equation has base of operations 10, use the common logarithm.
- If none of the terms in the equation has base of operations ten, utilise the natural logarithm.
- Use the rules of logarithms to solve for the unknown.
Think Near Information technology
Is in that location whatever way to solve [latex]{2}^{x}={3}^{ten}[/latex]?
Apply the textbox below to formulate an answer or example before you look at the solution.
Show Answer
Yes. The solution is ten = 0.
Equations Containing [latex]due east[/latex]
Case
Solve [latex]100=20{e}^{2t}[/latex].
Inapplicable Solutions
Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is right algebraically but does not satisfy the conditions of the original equation. I such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, think that the argument of the logarithm must exist positive. If the number we are evaluating in a logarithm function is negative, there is no output.
In the next instance we volition solve an exponential equation that is quadratic in grade. We will cistron first and and then use the nil product principle. Note how we find two solutions, but decline 1 that does not satisfy the original equaiton.
Case
Solve [latex]{east}^{2x}-{due east}^{x}=56[/latex].
Assay of the Solution
We tin use the one-to-ane belongings of exponents to solve exponential equations whose bases are the aforementioned. The terms in some exponential equations tin be rewritten with the same base, assuasive us to use the same principle. At that place are exponential equations that do not have solutions because we define exponential functions as having a positive base of operations. When restrictions are placed on the inputs of a office, it is natural that there will be restrictions on the output as well.
The inverse functioning of exponentiation is the logarithm, so we can use logarithms to solve exponential equations whose terms do non take the same bases. This is similar to using multiplication to "undo" division or improver to "disengage" subtraction. It is of import to bank check exponential equations for extraneous solutions or no solutions.
Put it Together
Ground Move recorded on seismic station at Penn Country, University Park Campus (Deike Bldg)
In the outset of this module nosotros left Joan with a plan to try and stump her granddaddy, the geologist, with the following math trouble.
The Alaska quake of 1964 had a Richter calibration value of viii.v, how many times greater than baseline was the measured wave amplitude recorded by a seismograph?
[latex]8.v=\mathrm{log}\left(\frac{A}{A_{0}}\right)[/latex]
He was and then excited to accept found something he and Joan could talk about, he beamed when he saw that she had a problem for him to solve.
"I probably knew how to solve that once, but information technology has been a long fourth dimension," said Grandfather. "Volition y'all show me?"
First, Joan explained to her grandpa that the goal of the problem was to go the statement out of the logarithm. She remembered that a logarithm is an exponent, so her first footstep was to rewrite the logarithm as an exponential:
[latex]\begin{array}{c}eight.5=\mathrm{log}\left(\frac{A}{A_{0}}\right)\\{10}^{8.v}=\frac{A}{A_{0}}\\{10}^{8.5}\cdot{A_{0}}={A}\stop{assortment}[/latex]
Joan entered [latex]{x}^{8.5}[/latex] into a calculator and got the following number:
[latex]316,227,766.017[/latex]
Wow, the aamplitude of the waves that caused the Alaska earthquake of 1964 was iii million times greater than the baseline reading!
When working with values that are very big or very pocket-sized, information technology is very helpful to piece of work in logarithmic or exponential scales. Information technology helps scientists avoid having to do calculations with massive numbers. Additionally, it makes comparisons between different measurements easier to understand.
Joan had such a adept fourth dimension explaining to Granddad how the math problem worked, she forgot all about having stumped him. Subsequently that night, Hobbes didn't wake her upwardly as he let himself out of the window and down the ladder Anne and Joan made for him. And Joan was happily saving abroad for the graduation trip that she was planning to take with Hazel (and perchance even her adjacent dreamy boyfriend). Algebra kept Joan from driving while intoxicated, taught her how food poisoning worked, and even helped her choose a expert cell telephone plan. But this isn't what Joan will think now that she'south finished her math form. What she'll remember is that she now has the ability and knowledge to effigy out solutions to so many of the challenges in her real life. And we hope you retrieve that, too! (Don't worry; if you forget some of the math rules forth the manner, yous tin ever bank check back and refresh your retention. We'll be waiting.)
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Source: https://courses.lumenlearning.com/coreq-mathforliberalarts/chapter/exponential-and-logarithmic-equations/
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