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# Introduction to Exponential Equations

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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 = 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.

Terms to Know
Exponential Equation

an equation involving a constant value raised to a variable power