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$\log(10 \times\sqrt{x-3})$

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Properties of logarithms: mixed review

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Q.

\log(10 \times\sqrt{x-3})

Question Prompt: Question prompt: What is the value of $\log(10 \times \sqrt{x - 3})$ ?
Understand the Problem: Understand the problem. $\newline$ We need to evaluate the expression $\log(10 \times \sqrt{x - 3})$ . This is a logarithmic expression, and we cannot simplify it further without knowing the value of $x$ . However, we can express it in a simpler logarithmic form using logarithmic properties.
Apply Logarithmic Property: Apply the logarithmic property of the product. $\newline$ The logarithm of a product is equal to the sum of the logarithms of the individual factors. Therefore, we can write $\log(10 \times \sqrt{x - 3})$ as $\log(10) + \log(\sqrt{x - 3})$ .
Simplify First Term: Simplify the first term. $\newline$ Since $\log(10)$ is a common logarithm (base $10$ ), and we are taking the log of the base itself, $\log(10)$ simplifies to $1$ .
Apply Square Root Property: Apply the logarithmic property of the square root. The square root can be written as an exponent of $\frac{1}{2}$ . Therefore, $\log(\sqrt{x - 3})$ can be written as $\log((x - 3)^{\frac{1}{2}})$ .
Apply Exponent Property: Apply the logarithmic property of exponents. $\newline$ The logarithm of a power is equal to the exponent times the logarithm of the base. Therefore, $\log((x - 3)^{\frac{1}{2}})$ can be written as $\frac{1}{2} \cdot \log(x - 3)$ .
Combine Simplified Terms: Combine the simplified terms. $\newline$ Now we combine the results from Step $3$ and Step $5$ to get the final simplified expression: $1 + (\frac{1}{2}) \cdot \log(x - 3)$ .

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