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Solving Equations Involving Logarithmic and Exponential Functions
On completion of this module you will be able to: convert logarithmic with bases other than 10 or e use the inverse property of exponential and logarithmic functions to simplify equations understand the properties of logarithms use the properties of logarithms to simplify equations solve exponential and logarithmic equations
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Bases other than 10 or e Most calculators have log x (base 10) and ln x (base e). How can we solve equations involving bases other than 10 or e? One way is using the change of base rule:
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Example
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Answer
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Answer
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Using the inverse property
When an exponential function and a logarithmic function have the same base, they are inverses and so effectively cancel each other out. Example Solve for x:
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We can’t divide by log! Use the exponential function with the same base (10) – called taking the anti-log. The left and right sides of the equation become exponents with a base of 10
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Example Solve for x: Answer Quick solution is to rearrange using the definition of logs: Alternative: Now rearrange to isolate the variable:
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Take anti-logs:
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Example Solve: Answer We have an exponential function (base e) which we can cancel out by taking the logarithm with the same base (ln x):
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Properties of logarithms
Example 1 (Since 84 = 12 7)
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Example 2 Solve for x: Answer
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Note that although both +7. 0711 and -7
Note that although both and square to give 50, only solves the original equation. Check: as required, but is undefined. Always check that your answer solves the original problem!!
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Example or
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Example
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Example Note Rules 1 to 4 have been expressed in base 10, but are equally valid using any base…
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This rule also works for any base e.g.
since
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Rule 6 also extends to other bases.
Whenever we take the log of the same number as the base, then the answer is 1. e.g.
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Let’s use Rules 3 and 6 to show why Rule 7 is true.
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This uses the concept of log and exponential functions as inverses as we discussed earlier.
This rule also works for other bases.
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Recall that this is the change of base formula used earlier.
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Example
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Example
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Example
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Summary: Rules of Logarithms
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Exponential and logarithmic equations
In solving equations which involve exponential and logarithmic terms, the following properties allow us to remove such terms and so simplify the equation.
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Example
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Example
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Example It doesn’t matter whether base 10 or base e is used, the result will be the same. Base Base e The numbers are different but the result is the same.
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Example The demand equation for a consumer product is
Solve for p and express your answer in terms of common logarithms. Evaluate p to two decimal places when q=60.
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Answer
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Example Suppose that the daily output of units of a new product on the tth day of a production run is given by: Such an equation is called a learning equation and indicates as time progresses, output per day will increase. This may be due to a gain in a worker’s proficiency at his or her job.
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Example (continued) Determine, to the nearest complete unit, the output on (a) the first day and (b) the tenth day after the start of a production run. (c) After how many days will a daily production run of 400 units be reached? Give your answer to the nearest day.
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Answer On the first day or production, t =1, so the daily output will be When t =10, Note that since the answers to parts (a) and (b) are the number of units of a new product, we have rounded these to the nearest whole unit.
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The production run will reach 400 units when q =400 or at
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Notice that the question requires the answer to be rounded to the nearest whole day.
If the answer were round to 8 days, so production has not quite reached 400. For this reason we round the answer to 9 days, even though production will be well passed 400 by the end of the 9th day.
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As always, we must check that the mathematically obtained solution answers the original question.
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