Exponential and Logarithmic Functions (58/36) -- Intermediate Algebra

Exponential and Logarithmic Functions Evaluate and Graph Logarithmic Functions Learning Objectives By the end of this section, you will be able to: - Convert between exponential and logarithmic form - Evaluate logarithmic functions - Graph Logarithmic functions - Solve logarithmic equations - Use logarithmic models in applications Before you get started, take this readiness quiz. We have spent some time finding the inverse of many functions. It works well to ‘undo’ an operation with another operation. Subtracting ‘undoes’ addition, multiplication ‘undoes’ division, taking the square root ‘undoes’ squaring. As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem. To deal with this we define the logarithm function with base a to be the inverse of the exponential function We use the notation and say the inverse function of the exponential function is the logarithmic function. The function is the logarithmic function with base , where and Convert Between Exponential and Logarithmic Form Since the equations and are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See (Figure). Notice the positions of the exponent and base. If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.” Convert to logarithmic form: ⓐ ⓑ and ⓒ Convert to logarithmic form: ⓐ ⓑ ⓒ ⓐ ⓑⓒ Convert to logarithmic form: ⓐ ⓑ ⓒ ⓐ ⓑⓒ In the next example we do the reverse—convert logarithmic form to exponential form. Convert to exponential form: ⓐ ⓑ and ⓒ Convert to exponential form: ⓐ ⓑ ⓒ ⓐ ⓑⓒ Convert to exponential form: ⓐ ⓑ ⓒ ⓐⓑ ⓒ Evaluate Logarithmic Functions We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation. Find the value of x: ⓐ ⓑ and ⓒ ⓐ ⓑ ⓒ *** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ & & & \hfill \phantom{\rule{1em}{0ex}}{\text{log}}_{\frac{1}{2}}\frac{1}{8}& =\hfill & x\hfill \\ \text{Convert to exponential form.}\hfill & & & \hfill {\left(\frac{1}{2}\right)}^{x}& =\hfill & \frac{1}{8}\hfill \\ \text{Rewrite}\phantom{\rule{0.2em}{0ex}}\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}{\left(\frac{1}{2}\right)}^{3}.\hfill & & & \hfill {\left(\frac{1}{2}\right)}^{x}& =\hfill & {\left(\frac{1}{2}\right)}^{3}\hfill \\ \text{With the same base, the exponents must be equal.}\hfill & & & \hfill x& =\hfill & 3\hfill & \text{Therefore,}\phantom{\rule{0.2em}{0ex}}{\text{log}}_{\frac{1}{2}}\frac{1}{8}=3\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} Extra alignment tab