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

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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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:

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

Example Solve for x: Answer Quick solution is to rearrange using the definition of logs: Alternative: Now rearrange to isolate the variable:

Take anti-logs:

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):

Properties of logarithms Example 1 (Since 84 = 12  7)

Example 2 Solve for x: Answer

Note that although both +7. 0711 and -7 Note that although both +7.0711 and -7.0711 square to give 50, only +7.0711 solves the original equation. Check: as required, but is undefined. Always check that your answer solves the original problem!!

Example or

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Example Note Rules 1 to 4 have been expressed in base 10, but are equally valid using any base…

This rule also works for any base e.g. since

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.

Let’s use Rules 3 and 6 to show why Rule 7 is true.

This uses the concept of log and exponential functions as inverses as we discussed earlier. This rule also works for other bases.

Recall that this is the change of base formula used earlier.

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Summary: Rules of Logarithms

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 It doesn’t matter whether base 10 or base e is used, the result will be the same. Base 10 Base e The numbers are different but the result is the same.

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.

Answer

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.

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.

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.

The production run will reach 400 units when q =400 or at

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.

As always, we must check that the mathematically obtained solution answers the original question.