Solve for x of an exponential function

In summary, the conversation discusses a problem involving the equation e^(2x+9) - 4e^x - 5 = 0 and suggests solving it by taking the logarithm of both sides and using the substitution u=e^x. The final solution is given as x = \ln u.
  • #1
sara_87
763
0

Homework Statement



e^(2x+9) - 4e^x -5 = 0

Homework Equations





The Attempt at a Solution


I changed this into: e^(2x+9) =4e^x + 5
I took logarithm of both sides:
2x+9=ln(4e^x +5)

but i don't know what to do with the right hand side.
what will be the easiest way to solve this?
Thank you very much
 
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  • #2
Try this,
[tex]\begin{align*}
0 & = e^{ 2 x + 9 } - 4 e^x - 5 \\
& = e^9 e^{ 2 x } - 4 e^x - 5.
\end{align*}[/tex]

Then the substitution [tex]u=e^x[/tex] turns this into a quadratic equation, which is easily solved, and then [tex]x = \ln u[/tex].
 
  • #3
excellent.
thank you v much
 

1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = abx, where a is the base and b is a constant. It is a type of function that grows or decays at a constant rate.

2. How do I solve for x in an exponential function?

To solve for x in an exponential function, you can use logarithms. Take the logarithm of both sides of the equation, and then use algebra to isolate x. Remember to check for any extraneous solutions, as some exponential equations may have more than one solution.

3. Can an exponential function have a negative base?

Yes, an exponential function can have a negative base. However, the base must be a real number and cannot be 0. Negative bases can result in negative outputs, which can be interpreted as decaying functions instead of growing functions.

4. What is the difference between an exponential function and a logarithmic function?

An exponential function involves raising a base to a variable power, while a logarithmic function involves taking the logarithm of a number to a specific base. In other words, exponential functions ask "what power do I raise the base to get this number?", while logarithmic functions ask "what power is this number when raised to the base?".

5. How are exponential functions used in the real world?

Exponential functions are used in many real-world applications, such as population growth, compound interest, and radioactive decay. They can also be used to model growth or decay in natural phenomena, such as bacteria growth or carbon dating. Exponential functions are also used in fields like economics, physics, and chemistry to describe various processes and relationships.

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