3 Tutorials that teach Introduction to Exponential Equations
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Introduction to Exponential Equations

Introduction to Exponential Equations


Introduction to Exponential Equations

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College Algebra

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  • Polynomial Equations versus Exponential Equations
  • Restrictions to the Base of Exponential Equations
  • Exponential Relationships in a Table

Polynomial Equations versus Exponential Equations

You may be familiar with equations that contain exponents. Take the polynomial equation y = 2x2 + 5x – 7 for example.  We see variables, and we see exponents.  However, this is not an exponential equation.  An exponential equation contains a variable exponent. This means that our variable x, for example, is part of the expression for the exponent, not a base number. 

Generally, we say that an exponential equation is given by: 

Exponential equation: an equation involving a constant value raised to a variable power.

We have a base number, b, being raised to a variable exponent, x.  We also have an a-value in front, which is a scalar multiplier to the exponential expression. 

Restrictions to the Base

The domain (or values that x is allowed to take on) is all real numbers, from negative infinity to positive infinity. As such, there are some restrictions to the base.  These are values for b that come into direct conflict with the all real number domain:

  • The base must not be negative.  This is because certain values of x would be excluded from the domain.  We can think of raising –5 to the power of 1/2.  This results in a non-real number (we can equivalently think of this as the square root of negative 5). 
  • The base must also not be zero.  This is because it excludes all negative values from x.  Take for example, 0 raised to the power of –2.  This can be thought of as 1 divided by 02 which is division by zero.  
  • The base cannot equal 1.  This doesn't represent a restriction to the domain, but if the base were 1, we actually wouldn't have an exponential relationship at all.  Rather, the relationship would be linear. 
  • In sum, b > 0, b ≠ 1

Exponential Relationships in Tables

Let's look at a table of values for the exponential equation y = 3 • 2x

When filling out values in a table, we apply the exponent first, and then multiply by the outside factor, as shown below:

In exponential equations, the exponent is applied to the base number first, and then multiplied by the scalar value in front of the exponential expression.