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Logarithmic Expression Evaluation
In this article, we will evaluate the logarithmic expression:
$2\log_{10}(8) + \log_{10}(36) - \log_{10}(1.5) - 3\log_{10}(2)$
To evaluate this expression, we will use the properties of logarithms and the rules of logarithmic operations.
Step 1: Simplify the Expression
Using the power rule of logarithms, we can rewrite the expression as:
$2\log_{10}(2^3) + \log_{10}(6^2) - \log_{10}(1.5) - 3\log_{10}(2)$
Step 2: Evaluate the Logarithms
Using the definition of logarithms, we can evaluate each term as follows:
$\log_{10}(2^3) = 3\log_{10}(2)$
$\log_{10}(6^2) = 2\log_{10}(6)$
$\log_{10}(1.5) = \log_{10}\left(\frac{3}{2}\right) = \log_{10}(3) - \log_{10}(2)$
Substituting these values into the original expression, we get:
$2(3\log_{10}(2)) + 2\log_{10}(6) - (\log_{10}(3) - \log_{10}(2)) - 3\log_{10}(2)$
Step 3: Combine Like Terms
Combining like terms, we get:
$6\log_{10}(2) + 2\log_{10}(6) - \log_{10}(3) + \log_{10}(2) - 3\log_{10}(2)$
Simplifying further, we get:
$3\log_{10}(2) + 2\log_{10}(6) - \log_{10}(3)$
Thus, the final simplified expression is:
$\boxed{3\log_{10}(2) + 2\log_{10}(6) - \log_{10}(3)}$
