How do I solve exponentials with base e?

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In summary: Therefore, the solution to the equation e^x=20 can be found by taking the natural log of both sides, resulting in x=ln(20). As for the second problem, since it cannot be solved explicitly, other methods such as graphing or iteration can be used to approximate the solutions. In summary, the conversation discusses the problem of solving equations involving exponential and logarithmic functions. It is mentioned that this type of problem cannot always be solved exactly and may require alternative methods such as graphing or iteration. The conversation also touches on the inverse relationship between exponential and logarithmic functions, and the use of logarithmic rules to solve equations.
  • #1
LinkMage
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I don't know how to solve x for things like e^x=20.
The problem I have to solve is x^3+e^(2x)+8=0
Can anyone help, please?
 
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  • #2
have you learned logarithms?

what is the natural log of e^x?
 
  • #3
LinkMage said:
I don't know how to solve x for things like e^x=20.
The problem I have to solve is x^3+e^(2x)+8=0
Can anyone help, please?

You won't be able to find an exact solution to your problem so you will have to resort to solving it graphically or by iteration.
 
  • #4
The exponential function is the inverse function of the natural logarithm.
therefore [tex] ln(e^{f(x)}) = f(x)[/tex]. In your example, like any equation, what you do on one side of the equation must be done on the other side. You can take the natural log of both sides [tex]ln(e^{x})=ln(20) [/tex]
Then using the property I gave above you should be able to solve for x

Once you have practised using the idea given above, you can then tackle your second question. You may want to bring your x-terms on one side of equation and any other terms, to the other side of equation. As with the first query, take the natural log of both sides. Though you eliminate your exponential function, you may still have some natural log terms left. That's okay, by substitution of values for x you should arrive at correct value f(x) to your question.

Edit: By plotting the values of x you substitute and resulting values f(x), as Tide suggests; by observing the trend in your graph, you may find (or at least narrow down) your solutions more quickly.
 
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  • #5
You mean I just replace values for x until I get that the equation equals 0. That's what I did first and got aproximately -2.001. I was just wondering if I could solve the equation instead of replacing numbers.
 
  • #6
Anytime you have a logarithm, there are different rules that you need to know. For the first example you gave the one you need to know is this.

[tex]\log_{a}b^c=c\log_{a}b[/tex] it also follows that

[tex]\ln a^b=b\ln a[/tex]

So when you are given something like [tex]e^x=20[/tex], how can you apply this rule to solve that?

For your second problem, as others have said, you cannot solve for x explicity so another method must be used.
 
  • #7
zwtipp05 said:
have you learned logarithms?

what is the natural log of e^x?
it is equal to x,

well, I just wanted to remind theOP
 

1. How do I simplify an exponential expression with base e?

To simplify an exponential expression with base e, you can use the natural logarithm function (ln). The natural logarithm of a number is the power to which e would have to be raised to equal that number. For example, ln(e) = 1 and ln(e2) = 2. To solve an expression like ex, you would use the property that ln(ex) = x.

2. What is the difference between e and other bases in exponential expressions?

The value of e is approximately 2.71828, and it is the base of the natural logarithm function. Unlike other bases, the value of e is a constant and is used in many mathematical and scientific calculations. It is also a special number in calculus and has many applications in the fields of physics and engineering.

3. How do I solve an exponential equation with base e?

To solve an exponential equation with base e, you can use the natural logarithm function (ln). For example, if you have an equation like ex = 10, you would take the natural logarithm of both sides to get ln(ex) = ln(10). Then, using the property mentioned in the first question, you can simplify it to x = ln(10). This method can be used to solve any exponential equation with base e.

4. Can I convert an exponential expression with base e to a logarithmic expression?

Yes, you can convert an exponential expression with base e to a logarithmic expression. Using the definition of logarithms, you can rewrite ex = y as ln(y) = x. This means that the logarithm of the value y with base e is equal to the exponent x. This is a useful tool for solving equations with e as the base.

5. How do I solve an exponential expression with a negative exponent?

To solve an exponential expression with a negative exponent, you can use the property that any number raised to a negative exponent is equal to its reciprocal raised to the positive version of that exponent. For example, e-2 = 1/e2. This can be rewritten as 1/e2 = e-2. Then, you can use the rules for exponentials with base e to simplify further.

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