Many logarithms may be calculated by converting them to exponential form. Suppose we want to calculate the value of log. log 16?

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Lauren Jones
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1 Question : How do you evaluate a arithm? Many arithms may be calculated by converting them to exponential form. Suppose we want to calculate the value of arithmic form, 16. Start by writing this expression as a 16? We could write the output with a variable, but a question mark suffices to indicate what we want to find. If we convert this form to an exponential form with a base of,? 16 The left hand side may be written with the base as place of 16,? 4 4. Substitute this expression in Since the exponent on the left side must be 4, this is also the value in the original exponential form, 16 4 This strategy works well as long as we can write the number on the right with the same base as the exponential on the other side of the equation. Example 4 Evaluate the Logarithm Find the value of each arithm by converting to exponential form. a. 81 Solution Write the arithm in arithmic form and convert to exponential form,? 81? means that 81

2 Since 81, the value of the arithm is, b. 3 Solution The arothmic and exponential forms are 1? 1 3? means that 3 Since , the value of the arithm is -3, 3 3. The negative power makes the reciprocal. c. ln e Solution The arothmic and exponential forms are ln e? means that e e? The value of the arithm is, ln e. In effect, the number is put into the base of e and the natural arithm reverses this process. Not every arithm may be solved by converting to exponential form. For this strategy to work, we must be able to write each side of the exponential form with the same base. Scientific and graphing calculators are both able to calculate natural arithms and common arithms. Natural arithms are calculated using a button labeled something like LN. Using this button, you should be able to do the following calculations by pressing the LN button, entering the number, and pressing the ENTER or = button. 8

3 ln 1 0 ln ln You may also calculate common arithms in a similar manner using a button that is typically labeled LOG. Using this button, you should be able to compute each of the following common arithms Using these buttons, you can compute any natural or common. Even the s that may not be solved by converting to exponential form may be computed on a calculator. Some calculator may even have a button for calculating a arithm with any positive base. To see if your calculator is able to do this, consult the manual for your calculator. If your calculator does not have this button, you can use the change of base formula to compute arithms with any positive base. Change of Base Formula for Logarithms For any positive base a and b not equal to 1, a x b b x a where x is a positive number. This formula is used to compute a arithm with base a by converting it to two arithms with base b. The base b can be any positive number not equal to 1, but usually it is a base of 10 or e so that a calculator may be used to compute the right hand side of the formula.

4 Example 5 Compute the Logarithm Find the value of each arithm using the Change of Base formula for Logarithms. a. 10 Solution We may use the Change of Base formula to convert this arithm to natural arithms or common arithms. If we convert to natural arithms we get 10 ln 10 ln A calculator is used to evaluate the natural arithms. The values of the individual arithms are shown above, but it is a good idea to type the entire expression. This avoids rounding in the middle of the problem and then rounding again at the end. Ideally you should only round once. If we convert to common arithms, The value of the original base arithm is the same whether it is computed from natural s or common s. b

5 Solution Use the Change of Base formula with natural arithms to give ln 100 ln You may also calculate the value using common arithms,

Section 1.5. Finding Linear Equations

Section 1.5. Finding Linear Equations Section 1.5 Finding Linear Equations Using Slope and a Point to Find an Equation of a Line Example Find an equation of a line that has slope m = 3 and contains the point (2, 5). Solution Substitute m =