Exponential Equations: Introduction and Simple Equations

An exponential function has the form:

EXPONENTIAL FUNCTION

y = ab^x

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

Some examples are:

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

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

3. y = 25,000 x 0.25^x (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: b^x = b^y if and only if x = y One-to One Property

Property 4: log_b b^x = 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 = 8^x

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 b^x = b^y. In other words rewrite 4096 as an exponent with base 8.

8⁴ = 8^x

Step 3: Solve for x.

Property 3 states that b^x = b^y if and only if x = y, therefore 4 = x.

4 = x

Example 1:

{(\frac{1}{4})}^{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 b^x = b^y. In other words rewrite 16 as an exponent with base 4.

${(\frac{1}{4})}^{x} = 16$

4^-x = 16

4^-x = 4²

Step 3: Solve for x.

Property 3 states that b^x = b^y if and only if x = y, therefore -x = 2

-x = 2

x = -2

Example 2: 14^x = 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.

$l o g_{14} 14^{x} = l o g_{14} 5$

Step 3: Solve for x

Property 4 states that log_bb^x = x, therefore the left-hand side becomes x.

$x = l o g_{14} 5$

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