# Exponential Equations: Introduction and Simple Equations

An exponential function has the form:

EXPONENTIAL FUNCTION

y = abx

Where a ≠ 0, the base b ≠ 1 and x is any real number

Some examples are:

1. y = 3x (Where a = 1 and b = 3)

2. y = 100 x 1.5x (Where a = 100 and b = 1.5)

3. y = 25,000 x 0.25x (Where a = 25,000 and b = 0.25)

When b > 1, as in examples 1 and 2, the function represents exponential growth as in population growth. When 0 < b < 1, as in example 3, the function represents exponential decay as in radioactive decay.

Some basic properties of exponential functions are:

Property 1: b0 = 1

Property 2: b1 = b

Property 3: bx = by if and only if x = y     One-to One Property

Property 4: logb bx = x           Inverse Property

Just as division is the inverse function to multiplication, logarithms are inverse functions to exponents. This is shown in Property 4.

Let's solve some simple exponential equations:

4096 = 8x
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since 4096 can be written as an exponent with base 8, this property is most appropriate. Property 3 - One to One Step 2: Apply the Property. To apply Property 3, first rewrite the equation in the form of bx = by. In other words rewrite 4096 as an exponent with base 8. 84 = 8x Step 3: Solve for x. Property 3 states that bx = by if and only if x = y, therefore 4 = x. 4 = x
Example 1:     ${\left(\frac{1}{4}\right)}^{x}=16\to {4}^{-x}=16$
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since 16 can be written as an exponent with base 4, Property 3 is most appropriate. Property 3 - One to One Step 2: Apply the Property. To apply Property 3, first rewrite the equation in the form of bx = by. In other words rewrite 16 as an exponent with base 4. ${\left(\frac{1}{4}\right)}^{x}=16$ 4-x = 16 4-x = 42 Step 3: Solve for x. Property 3 states that bx = by if and only if x = y, therefore -x = 2 -x = 2 x = -2
Example 2:      14x = 5
 Step 1: Choose the most appropriate property. Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since 14 cannot be written as an exponent with base 5, Property 3 is not appropriate. However the x on the left-hand side of the equation can be isolated using Property 4. Property 4 - Inverse Step 2: Apply the Property. To apply Property 4, take the log with the same base as the exponent of both sides. Since the exponent has a base of 14 then take log14 of both sides. $lo{g}_{14}{14}^{x}=lo{g}_{14}5$ Step 3: Solve for x Property 4 states that logbbx = x, therefore the left-hand side becomes x. $x=lo{g}_{14}5$

 Related Links: Math algebra Exponential Equations: Simple Equations with the Natural Base Natural Exponential Equations - Complex Equations Algebra Topics Exponential Functions

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