How can logarithms be used to solve exponential equations?

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In summary, to solve the equation 2^{k-2}=3^{k+1}, you can use the fact that \frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n and the inverse function of the exponential, which is the logarithm. This will help you simplify the equation and solve for k.
  • #1
anonymous12
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Homework Statement


Solve [itex]2^{k-2}=3^{k+1}[/itex]



Homework Equations





The Attempt at a Solution


[tex]2^{k-2}=3^{k+1}[/tex]
[tex]\frac{2^{k}}{2^{2}}=(3^k)(3^1)[/tex]
[tex]\frac{2^{k}}{3^{k}} = 4 \cdot 3[/tex]
[tex]\frac{2^{k}}{3^{k}} = 12[/tex]

What do I do next to solve for K?
 
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  • #2
Use the fact that [tex]\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n[/tex] and also, what do you know about logarithms?
 
  • #3
You could have used logarithms right from the start- [itex]log(2^{k-2})= log(3^{k+1})[/itex].

In general, to solve an equation of the form f(x)= constant or f(p(x))= f(q(x)) you will need to use the inverse function to f. And the inverse of the exponential is the logarithm.
 

What is an exponential equation?

An exponential equation is an equation in which the variable appears in the exponent. It can be written in the form y = ab^x, where a and b are constants and x is the variable.

How do you solve an exponential equation?

To solve an exponential equation, you need to isolate the variable on one side of the equation. This can be done by taking the logarithm of both sides or by using the properties of exponents. Then, you can solve for the variable using algebraic techniques.

What are the common methods for solving exponential equations?

The most common methods for solving exponential equations are taking the logarithm of both sides, using the properties of exponents, and graphing the equation to find the intersection point with the x-axis. Depending on the equation, other methods such as substitution or factoring may also be used.

Can an exponential equation have more than one solution?

Yes, an exponential equation can have more than one solution. This is because exponential functions are one-to-one, meaning that each input has a unique output. Therefore, it is possible for the same output to be produced by different inputs, resulting in multiple solutions.

What are some real-life applications of solving exponential equations?

Exponential equations are used in various fields such as finance, population growth, and radioactive decay. They can also be used to model the growth or decay of natural phenomena, such as bacteria growth, interest on investments, and the spread of diseases.

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