What is the formula for an exponential function?
Remember that our original exponential formula was y = abx. You will notice that in these new growth and decay functions, the b value (growth factor) has been replaced either by (1 + r) or by (1 - r). The growth "rate" (r) is determined as b = 1 + r. The decay "rate" (r) is determined as b = 1 - r.
An exponential function is a mathematical function of the following form: f ( x ) = a x. where x is a variable, and a is a constant called the base of the function. The most commonly encountered exponential-function base is the transcendental number e , which is equal to approximately 2.71828.
- On most calculators, you enter the base, press the exponent key and enter the exponent. Here's an example: Enter 10, press the exponent key, then press 5 and enter. (10^5=) The calculator should display the number 100,000, because that's equal to 105.
- Apply the growth rate formula. Simply insert your past and present values into the following formula: (Present) - (Past) / (Past) . You'll get a fraction as an answer - divide this fraction to get a decimal value. Express your decimal answer as a percentage.
- Graphs of Exponential Functions. The graph of y=2x is shown to the right. Here are some properties of the exponential function when the base is greater than 1. The graph passes through the point (0,1) The domain is all real numbers.
Exponential Equations. An exponential equation is one in which a variable occurs in the exponent, for example, . When both sides of the equation have the same base, the exponents on either side are equal by the property if , then .
- Biological exponential growth is the exponential growth of biological organisms. When the resources availability is unlimited in the habitat, the population of an organism living in the habitat grows in an exponential or geometric fashion.
- Graphs. A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.
- Exponential Expressions. The use of an exponential is a very convenient way of expressing the repeated multiplication of a number by itself. The exponent is placed to the upper right of the base number and signifies how many times the base term is multiplied by itself.
Method 1 Solving Basic Exponents
- Learn the correct words and vocab for exponent problems.
- Multiply the base repeatedly for the number of factors represented by the exponent.
- Solve an expression: Multiply the first two numbers to get the product.
- Multiply that answer to your first pair (16 here) by the next number.
- e and ln cancel each other out leaving us with a quadratic equation. x = 0 is impossible as there is no way of writing 0 as a power. Put in the base number e. ln and e cancel each other out. Simplify the left by writing as one logarithm.
- Dividing is the inverse (opposite) of Multiplying. A negative exponent means how many times to divide by the number. Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125. Or many divides: Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008.
- A Review of the Rules for Exponents. Below is List of Rules for Exponents and an example or two of using each rule: Zero-Exponent Rule: a0 = 1, this says that anything raised to the zero power is 1. Power Rule (Powers to Powers): (am)n = amn, this says that to raise a power to a power you need to multiply the exponents
Updated: 4th October 2019