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ACC Module #3 Unit 3.5

ACC Module #3 Unit 3.5

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ACC Module #3 Unit 3.5

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  1. DemingEarly College High SchoolUnit 3.0 Advanced Algebra and Functions (AAF) 3.5 Exponential and Logarithmic Equations

  2. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations A logarithmic function is an inverse for an exponential function. The inverse of the base b exponential function is written as , and is called the base b logarithm. The domain of a logarithm is all positive real numbers. It has the properties that . For positive real values of x, . When there is no chance of confusion, the parentheses are sometimes skipped for logarithmic functions: may be written as . For the special number e, the base e logarithm is called the natural logarithm and is written as . Logarithms are one-to-one.

  3. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations 3.5.1 Solving Logarithmic and Exponential Functions To solve an equation involving exponential expressions, the goal is to isolate the exponential expression. Once this process is completed, the logarithm - with the base equaling the base of the exponent of both sides - needs to be taken to get an expression for the variable. If the base is e, the natural log of both sides needs to be taken. To solve an equation with logarithms, the given equation needs to be written in exponential form, using the fact that means , and then solved for the given variable. Lastly, properties of logarithms can be used to simplify more than one logarithmic expression into one.

  4. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations 3.5.1 Solving Logarithmic and Exponential Functions When working with logarithmic functions, it is important to remember the following properties. Each one can be derived from the definition of the logarithm as the inverse to an exponential function: [1][2][3] [4] [5] [6]

  5. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations 3.5.1 Solving Logarithmic and Exponential Functions When solving equations involving exponentials and logarithms, the following fact should be used: If f is a one-to-one function, is equivalent to . Using this, together with the fact that logarithms and exponentials are inverses, allows the manipulation of the equations to isolate the variable. For example, solve: . Using the definition of a logarithm, the equation can be changed to . The functions on the right side cancel with a result of . This then gives .

  6. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations 3.5.1 Solving Logarithmic and Exponential Functions Exponential expressions can also be rewritten, just as quadratic equations. Properties of exponents must be understood. Multiplying two exponential expressions with the same base involves adding the exponents. Dividing two exponential expressions with the same base involves subtracting the exponents. Raising an exponential expression to another exponent includes multiplying the exponents:

  7. Unit 3.0 Advanced Algebra and Functions (AAF)3.5 Exponential and Logarithmic Equations 3.5.1 Solving Logarithmic and Exponential Functions The zero power always gives a value of 1: . Raising either a product or a fraction to a power involves distributing that power: and Finally, raising a number to a negative exponent is equivalent to the reciprocal including the positive exponent:

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