Solving Exponential Equations

  • Thread starter froilan041
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In summary, an exponential equation is a mathematical expression in which the variable appears in an exponent. To solve such equations, one can use logarithmic properties and algebraic techniques. The key steps to solving exponential equations are isolating the exponential term, taking the logarithm of both sides, simplifying, solving for the variable, and checking the solution. Some common mistakes include not applying logarithmic properties correctly and confusing the order of operations. Exponential equations have various real-world applications, such as modeling population growth and analyzing data in fields like biology and economics. They are also used in signal processing, electronics, and communication systems.
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froilan041
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Can anyone show me how to solve the following equations:

2^(x-1)-2^(x)=2^(-3)

3^(x+1)+3^(x)=36

I would greatly appreciate your assistance.
 
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  • #2
2x-1=2x*2-1.

It should be a bit easier now.
 
  • #3
Here is a general way to solve things like this
1. [tex]2^{x-1}-2^{x} = 2^{-3}[/tex]
Now factor out a [tex]2^{x-1}[/tex]
[tex]2^{x-1}(1 - 2) = 2^{3} \rightarrow 2^{x - 1} = -\frac{1}{8}[/tex]
You can take the logarithm at this point or you could notice that [tex]-\frac{1}{8}
[/tex] is out of the range of [tex]2^{x}[/tex]

Similar approach
2. [tex]3^{x+1} + 3^{x} = 36 \rightarrow
3^{x}(3 + 1) = 36 \rightarrow
3^{x} = 9 \rightarrow
x = 2 [/tex]
 

What is an exponential equation?

An exponential equation is an equation in which the variable appears in an exponent, such as y = 3^x. It is a type of mathematical expression that involves repeated multiplication of a base number raised to a power.

How do you solve an exponential equation?

To solve an exponential equation, you can use the properties of logarithms, specifically the power rule, product rule, and quotient rule. You can also use the definition of logarithms to rewrite the equation in a more manageable form. Once you have the logarithms on both sides of the equation, you can solve for the variable using algebraic techniques.

What are the key steps to solving exponential equations?

The key steps to solving exponential equations are:

  1. Isolate the exponential term on one side of the equation.
  2. Take the logarithm of both sides using the same base.
  3. Use the properties of logarithms to simplify the equation.
  4. Solve for the variable using algebraic techniques.
  5. Check your solution by plugging it back into the original equation.

What are some common mistakes when solving exponential equations?

Some common mistakes when solving exponential equations include:

  • Forgetting to take the logarithm of both sides of the equation.
  • Using the wrong base for the logarithm.
  • Not applying the properties of logarithms correctly.
  • Forgetting to check the solution by plugging it back into the original equation.
  • Confusing the order of operations.

What are some real-world applications of exponential equations?

Exponential equations have numerous real-world applications, such as modeling population growth, radioactive decay, and compound interest. They can also be used to analyze data in fields such as biology, economics, and physics. Additionally, they are used in signal processing, electronics, and communication systems.

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