3
Tutorials that teach
Introduction to Exponential Equations

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Tutorial

- 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 = 2x^{2} + 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 0
^{2}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 • 2^{x}

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.