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# Class Note for MATH 1330 with Professor Flagg at UH

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15
PreCalSection32 Math 1330 Section 32 Logarithmic Functions De ning and Evaluating Logarithms Let x gt 0 For a gt 0 and a at l the logarithmic function with base a denoted by loga x is given by loga x y if and only if ay x Thus loga x is the exponent to which the base a must be raised to give x The common logarithm is the logarithm with base 10 and it is denoted by 10gx 10g10x The natural logarithm is the logarithm with base e and it is denoted by W log on Let s rst practice converting the logarithm notation to its equivalent exponential form Example 1 Write each equation in exponential form A logs l 0 l B lo 4 g16 C log327 3 Next let s try writing the logarithmic form of an exponential equation Example 2 Write the following equations in their logarithmic form A 341 9 B 257 5 PreCalSection32 Next we will use the equivalent exponential form of a logarithm to evaluate an expression Example 3 Simplify each expression A log1000 1 131 A C log9 3 Finally we can use the equivalent exponential form of a logarithm to solve simple logarithmic equations Example 4 Solve each equation for x by using the exponential form of the logarithm A log2 x 16 B log10000 x C log3 x 3 D logr 64 3 In the next section we will develop the logarithm rules that are consequences of the de nition of logarithm This will make some of these problems much faster but it is important to try some problems by the definition to see why the rules work Inverse Functions Now let s try simplifying loga ax PreCalSection32 Next simplify alga What does this mean The logarithm base a and exponential function base a are inverses of each other The functions f x log x and gx a are both oneto one functions and are inverses of each other Example 5 Simplify the following expressions A log22 B 71ng73 c 1115 D e1 Graphng Logarithmic Functions Let a be a positive real number With a 0 Then the function of the form fx logax is called a logarithmic function With base a and the shape ofits graph depends on Whether a gt1 or a lt 1 If a gt1 the graph looks like PreCalSection32 4 Domain 0 00 Range oooo Vertical asymptote x 0 Key Points 10 and 611 fx gt oo asx gt0 If a lt1 the graph of fx loga x looks like Domain 0 00 Range oooo Vertical asymptote x 0 Key Points 10 and 611 fx gtooasx gt0 Graphing using Transformations Example 6 Sketch a graph of the function f x log2 x 1 1 State the domain range and asymptote of the function PreCalSection32 Example 7 Sketch a graph of the function gx log3 x 2 State the domain range and asymptote of the function Example 8 State the domains for the following functions A gx log2 4x 7 B fx ln x C fx log5 x2 l

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