Exponential Equations: Introduction and Simple Equations
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 2: b^{1} = b
Property 3: b^{x} = b^{y} if and only if x = y Oneto 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:
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. 
8^{4} = 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 
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. 
${\left(\frac{1}{4}\right)}^{x}=16$ 4^{x} = 16 4^{x} = 4^{2} 
Step 3: Solve for x.

x = 2 x = 2 
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 lefthand 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 log_{14} of both sides. 
$lo{g}_{14}{14}^{x}=lo{g}_{14}5$ 
Step 3: Solve for x Property 4 states that log_{b}b^{x} = x, therefore the lefthand 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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