Name : ______ ( ) Class : Date :_ Objectives: Unit 7: Logarithmic and Exponential Functions Graphs Solving Equations of the Form.

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Name : ______________ ( ) Class : ________ Date :_________ Objectives: Unit 7: Logarithmic and Exponential Functions Graphs Solving Equations of the Form Logarithmic Equations Laws of Logarithms Logarithms Common and Natural Logarithms

Graphs of Exponential Functions

Graph of y = lg x Graphs of Common Logarithms

Convert to logarithmic form. The logarithm or index for the given base is -2. Convert to index form. The base is 3. The base is 4. The logarithm or index for the given base is 3. Logarithms

If a logarithm is defined for base a, then and Special Cases Logarithms

Evaluate the following. Logarithms Example 1:

A common logarithm is a logarithm to the base 10. Tables of common logarithms were often used for calculating in the days before the electronic calculator. On a scientific calculator, common logarithms can be evaluated using the LOG key. Logarithms

A natural logarithm is a logarithm to the base e. Natural logarithms are also known as Naperian logarithms after John Napier ( ). On a scientific calculator, natural logarithms can be evaluated using the LN key. Logarithms

From the definitions of logarithms, the following statements are equivalent. Let’s use these definitions in some examples. and Logarithms

Convert the following to index form. Convert the following to logarithmic form. Index form Logarithmic form Logarithms Example 2:

Find y in terms of x. Index form Rearrange Alternate form Surds, Indices and Logarithms (a) (b) Example 3:

Solve for x. Evaluate using the calculator. Index form In most calculators, the function e x is on the same key as LN. Evaluate and solve for x. Logarithms Example 4:

Solve for x. Evaluate using the calculator. Index form In most calculators, the function 10 x is on the same key as LOG. Evaluate and solve for x. Logarithms Example 5:

The Power Law The Product Law The Quotient Law Logarithms

Let’s use these laws in some examples. The Change of Base Law A special case Logarithms

Example 6: Evaluate the following. Combine using the product and quotient laws. Apply the power law. Logarithms (a) (b)

Separate using the product and quotient laws. Apply the power law. Logarithms Example 7:

Combine, applying the quotient law. Rearrange and solve the equation. Find y in terms of x. Arrange the log terms on one side. Index form Logarithms Example 8:

Apply the power law. Apply the change of base law. Evaluate the following. Express as powers of 2, 5 and 10. Logarithms Example 9:

For two logarithms of the same base, Let’s use this property to solve some logarithmic equations. Logarithms An Important Property of Logarithms

Combine using the product law. Example : 10 Use the property of logarithms. Remember to check if the results are acceptable. Logarithms So, x = 3.

Apply the power law. Example 11: Solve the following equation. Index form Remember to check if the results are acceptable. Logarithms log 4 (6 – x) is defined for x = – 122.

Apply the change of base law. Example 12 : Solve the following equation. Substitute Both results are acceptable. Logarithms

For two logarithms of the same base, Let’s use this property to solve some logarithmic equations. Logarithms An Important Property of Logarithms

Combine using the product law. Example 13 : Use the property of logarithms. Remember to check if the results are acceptable. Logarithms So, x = 3.

Apply the power law. Example 14 : Solve the following equation. Index form Remember to check if the results are acceptable. Logarithms log 4 (6 – x) is defined for x = – 122.

Apply the change of base law. Example 15 : Solve the following equation. Substitute Both results are acceptable. Logarithms