Exponential functions - converting a linear equation

In summary, the problem involves finding the exponential function f1(t) = Ceλt given the values of t, yi, A1, and B1. By adding logarithms to both sides of the equation, a linear connection is formed between Yi and t, with coefficients A1 and B1. Further simplification leads to solving for the constants C and λ, using the values given for A1 and B1. This process can be applied to similar problems involving unknown exponential functions.
  • #1
LizzieL
12
0

Homework Statement


Ok, so I have an unknown exponential function:
y1 = f1(t)

By measuring the values of t and yi, a linear connection is generated between Yi (=log (yi)) and t:

Yi = Ait + Bi.
A1 = -2.12
B1 = 1.96

Problem 1: Describe f1 in the following matter:
y1 = f1(t) = Ceλt


The Attempt at a Solution


I've been juggling with numbers and letters without ending up with anything reasonable.
Ok, let's ramble down some thoughts:
Adding log function to both sides of the functions gives

log(y1) = log t + log -2.12 + log 1.96 = log (C) + log (e) λt

And this is where it all ends. Any help out there? Would be much appreciated!
 
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  • #2
LizzieL said:
Problem 1: Describe f1 in the following matter:
y1 = f1(t) = Ceλt


The Attempt at a Solution



Ok, let's ramble down some thoughts:
Adding log function to both sides of the functions gives

log(y1) = log t + log -2.12 + log 1.96 = log (C) + log (e) λt

The red part has no sense...

The logarithm of y is given: log(y)=At+B=-2.12 t + 1.96.

And -2.12 t + 1.96=log(e)(λt)+log(C)

What is the base of the logarithm?

Compare the terms. The constant terms have to be equal on both sides, and the coefficients of the variable t also have to be equal.

ehild
 
  • #3
Oh, of course the logarithm of y is given! Why didn't I see that before. Thanks for your reply.

The base of the logarithm is 10, forgot to say.
But, how do I solve it from here? This is my first go at a problem like this, so any help is appreciated. How can I solve this when I have no values for neither t, C and λ? and the log gets in the way.
 
  • #4
Welcome to PF, LizzieL! :smile:

ehild's equation must hold for any t.
This means that -2.12=log(e)(λ) and that 1.96=log(C).

For instance -2.12=log(e)(λ) can be rewritten as λ=-2.12/log(e).
 
  • #5
Oh my, you guys really are angels sent from up above. Thanks a whole lot! Also for the welcome :smile:

So, I have a similar problem:

log(y2) = 1.85t + 1.56 = log(e)μ(t-a)

I cannot use the same principle as before, right? Or can I?

If I try again...

1.85 = log(e)μ
1.56 = -log(e)μa

Is this way off?
 
  • #6
I'd say it is right on! :approve:
 

FAQ: Exponential functions - converting a linear equation

1. What are exponential functions?

An exponential function is a mathematical function in the form of y = ab^x, where a is the initial value, b is the growth factor, and x is the input variable. It is characterized by a constant rate of change, where the output increases or decreases at a faster and faster rate as x increases.

2. How do you convert a linear equation into an exponential function?

To convert a linear equation y = mx + b into an exponential function, you need to rewrite it as y = a(b^x), where a is the initial value and b is the growth factor. To find a and b, you can use the initial value and any other point on the linear equation to form a system of equations and solve for a and b.

3. What is the difference between a linear function and an exponential function?

The main difference between a linear function and an exponential function is their rate of change. A linear function has a constant rate of change, while an exponential function has an increasing rate of change. In other words, the output of a linear function increases or decreases by the same amount for every unit increase in the input, while the output of an exponential function increases or decreases by a larger amount for every unit increase in the input.

4. How do you graph an exponential function?

To graph an exponential function, you first need to plot a few points using the x and y values from the function. Then, you can connect the points with a smooth curve. Remember that exponential functions never touch the x-axis, but they can approach it as x approaches negative or positive infinity. It is also helpful to label the initial value, growth factor, and any other important points on the graph.

5. What are some real-life applications of exponential functions?

Exponential functions are commonly used to model growth or decay in many real-life situations, such as population growth, compound interest, radioactive decay, and bacterial growth. They are also used in fields like economics, physics, and biology to describe various natural phenomena and predict future outcomes.

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