Math Lesson 13.1.4 - The Combination of Logarithm Properties

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Welcome to our Math lesson on The Combination of Logarithm Properties, this is the fourth lesson of our suite of math lessons covering the topic of Definition and Properties of Logarithms, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Combination of Logarithm Properties

All the previously covered properties of logarithms are used to make operations and calculations easier. In the previous examples, we included only the corresponding property of logarithm, which we wanted to illustrate. However, in most cases, an exercise may require us to combine more than one of the above properties in order to obtain the simplest form or the result of the expression. Let's consider a couple of examples to illustrate this point.

Example 9

Write the following logarithmic expressions in the simplest form.

1. log₃ (9-1/3)² + 3 log_1/2 (4 + 1/2)⁴ -5 log₁6⁡√(16-1/4)³
2. 1/3 log₅ √1/625 + 2/3 log 3/30 + log₃ 18/log₃ 2 - 5^{log₅ 4}

Solution 9

1. Let's represent all operations in the following table, where all the properties used are shown on the right column.
2. Let's use the table method again for easier understanding. We have

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7. Continuing learning logarithms - read our next math tutorial: Exponential and Logarithmic Equations

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