Solving Exponential Equations: e^2x = 5e^3x

In summary, to solve the equation e^2x = 5e^3x, you need to take the natural log of both sides. Then, you can use the logarithm rule log(ab)=log(a)+log(b) to separate the constant 5 from the exponent. Alternatively, you can rewrite the equation as e^x=1/5 and continue solving from there.
  • #1
cue928
130
0
e^2x = 5e^3x


I understand that I need to take a natural log of both sides here, what I am thrown about is the constant "5". Can I bring that up as an exponent? So, e^2x = e^(3x)^5?
 
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  • #2
No, not really. That's a rule for logarithms, not for exponents.

However, it's not really a problem. Once you take the natural log of both sides, you just need to remember a different logarithm rule:

[tex]log(ab)=log(a)+log(b)[/tex]

That's all you need here.
 
  • #3
You might also rewrite this equation, dividing with 5e^2x on both sides:
e^x=1/5
 
  • #4
Sorry but I don't completely follow. I get that I need to take a natural log of both sides, so are you saying: ln(e^2x) = 5 ln(e^2x)?
 
  • #5
cue928 said:
Sorry but I don't completely follow. I get that I need to take a natural log of both sides, so are you saying: ln(e^2x) = 5 ln(e^2x)?

No, I'm saying that ln(e^2x)=ln(5e^(3x)). From there, you can use the logarithm rule I posted above to separate the 5.
 
  • #6
Char. Limit said:
No, I'm saying that ln(e^2x)=ln(5e^(3x)). From there, you can use the logarithm rule I posted above to separate the 5.

Thanks, got it!
 

What is an exponential equation?

An exponential equation is an equation in which the variable appears in the exponent. It typically takes the form of a^x = b, where a and b are constants and x is the variable.

How do I solve an exponential equation?

To solve an exponential equation, you can use the property of logarithms which states that log a^x = xlog a. This allows you to bring the variable down from the exponent, making it easier to solve.

How do I solve a exponential equation with different bases?

If the equation has the form a^x = b^x, you can take the logarithm of both sides using any base. If the equation has the form a^x = b, you can use the change of base formula to convert the bases to the same value, making it easier to solve.

What is the difference between an exponential equation and a logarithmic equation?

An exponential equation is in the form of a^x = b, where x is the variable and a and b are constants. A logarithmic equation is in the form log a^x = b, where a is the base, x is the exponent, and b is the result. In other words, exponential equations involve finding the value of the variable in the exponent, while logarithmic equations involve finding the value of the exponent.

How do I check my solution to an exponential equation?

To check your solution, you can substitute the value you found for the variable back into the original equation and see if it satisfies the equation. You can also use a graphing calculator to plot the original equation and your solution, and see if they intersect at the same point.

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