Introduction and Simple Equations
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 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:
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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 |
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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 |
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Step 3: Solve for x. Property 3 states that bx = by if and only if x = y, therefore 4 = x. |
4 = x |
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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 |
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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. |
4-x = 16 4-x = 42 |
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Step 3: Solve for x.
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-x = 2 x = -2 |
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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 |
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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. |
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Step 3: Solve for x Property 4 states that logbbx = x, therefore the left-hand side becomes x. |
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Related Links: Math algebra Simple Equations with the Natural Base Complex Equations with the Natural Base Algebra Topics |
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