Solution of exponential equation

  • Thread starter techiejan
  • Start date
  • Tags
    Exponential
In summary, the given equation can be solved for x by taking the square root of (2A/βB) when β is large. To do this, one must first expand e^(x/β) to the first three terms, as β is much larger than x and the other terms would have a negligible effect on the value of x. Including more terms in the expansion would result in a more accurate value of x.
  • #1
techiejan
8
0
How does the given equation:
βe^(x/β)-x = β+(A/B)

solves to x = √(2A/βB) when β is large?
 
Mathematics news on Phys.org
  • #2
Expand e[itex]^{\frac{x}{\beta}}[/itex] to the first three terms.
 
  • #3
Thanks grzz, I solved it now. Can you please tell me the logic behind expanding the exponential to first three terms?
 
  • #4
The expansion of e[itex]^{\frac{x}{\beta}}[/itex] consists of powers of [itex]\frac{x}{β}[/itex].

Since β is large (compared with x) then we can include only the first three terms of the expansion since the other terms would be very small and would not change the value of x.

Of course, if one wants a more accurate value of x, one must include more terms in the expansion.
 
  • #5
Thanks grzz. This was very helpful.
 

1. 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. Exponential equations are commonly used in science and mathematics to model growth and decay.

2. How do you solve an exponential equation with the same base?

To solve an exponential equation with the same base, you can use the property of logarithms that states that logb(x^a) = alogb(x). This means that you can take the logarithm of both sides of the equation and then solve for the variable.

3. What is the difference between solving an exponential equation and evaluating an exponential expression?

Solving an exponential equation means finding the value of the variable that makes the equation true. On the other hand, evaluating an exponential expression means simplifying the expression to a single numerical value. Solving an equation involves finding the value of the variable, while evaluating an expression does not involve any variables.

4. What are the common methods for solving exponential equations?

The most common methods for solving exponential equations include using logarithms, taking the nth root, and using the properties of exponents. The method used will depend on the specific form of the equation and the desired outcome.

5. Are there any real-world applications of exponential equations?

Yes, there are many real-world applications of exponential equations. They are commonly used in population growth and decay, compound interest, radioactive decay, and many other fields of science and economics. Exponential equations can also be used to model the spread of diseases, the growth of bacteria, and the depreciation of assets.

Similar threads

Replies
4
Views
116
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
778
Replies
4
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
970
  • General Math
Replies
1
Views
667
  • General Math
Replies
9
Views
2K
Replies
19
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
917
Back
Top