Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log √(100x)

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Recognize that the expression inside the logarithm is a square root, which can be rewritten as an exponent: \(\sqrt{100x} = (100x)^{\frac{1}{2}}\).

Use the logarithm power rule: \(\log(a^b) = b \log(a)\), to bring the exponent \(\frac{1}{2}\) in front of the logarithm: \(\log \sqrt{100x} = \frac{1}{2} \log(100x)\).

Apply the logarithm product rule: \(\log(ab) = \log a + \log b\), to separate the logarithm of the product inside: \(\log(100x) = \log 100 + \log x\).

Substitute back into the expression: \(\frac{1}{2} \log(100x) = \frac{1}{2} (\log 100 + \log x)\).

Evaluate \(\log 100\) by recognizing that \(100 = 10^2\), so \(\log 100 = 2\), and write the expanded form as \(\frac{1}{2} (2 + \log x)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, log(ab) = log a + log b and log(a^n) = n log a. Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Square Root as an Exponent

The square root of a number can be expressed as an exponent of 1/2. For example, √x = x^(1/2). This conversion is useful in logarithmic expressions because it allows the use of the power rule of logarithms to simplify or expand the expression.

Evaluating Logarithms Without a Calculator

Some logarithmic values can be evaluated exactly without a calculator, especially when the argument is a power of the logarithm's base. For example, log 100 with base 10 equals 2 because 10^2 = 100. Recognizing such values helps simplify expressions efficiently.

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