Solving Exponential Functions: A Comprehensive Guide with Examples

Exponential equations and functions are essential in many fields, including finance, biology, and physics. In this article, we will delve into the methods of solving exponential functions and provide step-by-step solutions for specific examples. By the end of this guide, you will be able to solve similar problems with ease.

Understanding Exponential Functions

An exponential function is a mathematical function in the form y a b^x, where a and b are constants, and x is the variable. The parameter b must be positive and b ≠ 1. Exponential growth occurs when b > 1, and exponential decay when 0 .

Example Problem 1: Finding the Value of an Exponential Function at a Specific Point

Let's solve the problem where we need to find the value of f_6 an exponential function given that f_5 29 and f_{15} 88.

Step-by-Step Solution

Set up the equations: We start with the general form of the exponential function f(x) a b^x. Given the points (5, 29) and (15, 88), we can write: 29 a b^5 88 a b^{15}

Step 2: Divide the second equation by the first:

frac{88}{29} frac{a b^{15}}{a b^5} frac{88}{29} b^{10}

Step 3: Solve for b:

b left(frac{88}{29}right)^{1/10}

Step 4: Substitute b back into the first equation to find a:

29 a left(left(frac{88}{29}right)^{1/10}right)^5 29 a left(frac{88}{29}right)^{1/2} a 29 cdot left(frac{29}{88}right)^{1/2}

Step 5: Calculate f_6:

f_6 a cdot b^6 f_6 29 cdot b^5 cdot b f_6 29 cdot 29 cdot b b left(frac{88}{29}right)^{1/10} f_6 approx 29 cdot left(frac{88}{29}right)^{1/10} f_6 approx 32.71

Example Problem 2: Solving for a Specific Value in an Exponential Function

Consider the function where f_{3.5} 16 and f_4 99. We need to find the value of f_6 to the nearest hundredth.

Step-by-Step Solution

Set up the logarithmic equations: We start with the natural logarithms of the given points: ln(f_{3.5}) ln(a) 3.5 ln(b) ln(16) ln(f_4) ln(a) 4 ln(b) ln(99) Subtract the first equation from the second: ln(99) - ln(16) (4 ln(b) - 3.5 ln(b)) lnleft(frac{99}{16}right) 0.5 ln(b) ln(b) 2 lnleft(frac{99}{16}right) b left(frac{99}{16}right)^2 f_6 a cdot b^6 f_6 99 cdot left(frac{99}{16}right)^{12} f_6 approx 145,109.57

Conclusion

In this article, we have covered the methods for solving exponential functions and provided detailed step-by-step solutions for specific examples. By understanding the properties of exponential functions and following these procedures, you can solve a wide range of problems related to exponential growth and decay.

Additional Resources

Wikipedia: Exponential Function Khan Academy: Exponential Equations Math is Fun: Exponential Growth