You can put this solution on YOUR website! Given to simplify:
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Notice that . Substitute for to change the expression to:
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By the rules of logarithms an exponent can be brought out as the multiplier of the log term. So bring the exponent 2 out and make it the multiplier of its log term. When you do that you have:
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The first two terms of the expression now involve . Let's do the same for the last term. Note that . So in the last term, replace by to get:
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By the rules of logarithms, the log of a product is equal to the sum of the logs of the two multipliers. So the last term can be changed as follows:
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Do the distributed multiplication of the last term by multiplying the 2 times each of the two log terms in the parentheses:
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The term means to what power do you have to raise 10 (the base) to make it equal 10? The answer to that is the power of 1. So is equal to 1. Substitute 1 for and the expression becomes:
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Do the multiplications of the constants in the second and last terms and the expression becomes:
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Add together the terms containing which is and the expression becomes:
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And that's the expression that results from simplifying the expression that you originally were given.
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Keep in mind that is just a number that you can look up on a calculator. If you enter 5 and press the log key, you will get the value 0.698970004. Substitute this for and the answer transforms to:
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Multiply out the first term and then add the 2 and it becomes:
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Which adds up to be:
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And this is the numerical solution to this problem.
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Hope this helps you to understand logarithm operations a little better.
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