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Creating a table of values for a function helps understand its behavior by displaying input-output pairs. In our case, we observe this for the function \( f(x) = \log x \), specifically in base 10.Let's consider the function values at specified points:

• \(x = 0.5\): Calculating \(\log_{10} 0.5\), we find \(f(0.5) \approx -0.30\). This means the value is negative as 0.5 is less than 1.
• \(x = 1\): \(\log_{10} 1 = 0\). Since logarithm is an exponent, this result confirms that 10 raised to the 0 power is 1.
• \(x = 2\): For \(\log_{10} 2\), we get \(f(2) \approx 0.30\). This signifies only a slight increase, since 2 is close to 1.
• \(x = 4\): Calculating \(\log_{10} 4\), the result is \(f(4) \approx 0.60\), illustrating gradual growth.
• \(x = 6\): For \(\log_{10} 6\), we compute \(f(6) \approx 0.78\), further demonstrating this increase.
• \(x=8\): \(\log_{10} 8\) gives \(f(8) \approx 0.90\), showing that as \(x\) increases, \(f(x)\) also goes up.
• \(x = 10\): As already known, \(\log_{10} 10 = 1\), confirming the property of logarithms where the base itself results in 1.

A useful tip: when filling in these values, always round to the nearest hundredth for precision and uniformity.

Graphing functions like \( f(x) = \log x \) paints a visual picture of the function's behavior. This aids in understanding the nature and the flow of mathematical relationships.When plotting the function using the table of values:- Start by surveying the points: \( (0.5, -0.30), (1, 0), (2, 0.30), (4, 0.60), (6, 0.78), (8, 0.90), (10, 1) \).- Remember, as \( x \) gets larger, \( f(x) \) increases but at a decreasing rate.- Notice the unique shape of a logarithm graph: the curve starts from the negative x-values and climbs upwards to the right.- The x-axis will have larger units compared to the y-axis which frequently has smaller increments, due to its slow increase.- Notice the asymptotic behavior as the curve approaches the y-axis but never touches it, reflecting the domain condition \( x > 0 \).This graphing approach reflects the true nature of logarithms, displaying how growth slows with larger x-values.

Logarithms are a concise way to express rates of change or growth in mathematical expressions. They are inverse operations to exponents.Key insights:

• The base 10 logarithm, \( \log_{10} \), signifies what power a base number must be raised to yield a particular number.
• The concept of \( \log_{10} x = y \) means \( 10^y = x \).
• For values \( x < 1 \), the logarithms are negative. This reflects numbers that produce fractions when base 10 is raised to a power.
• At \( x = 1 \), logarithms return zero, indicating any number raised to the zero power results in 1.
• As values of \( x \) increase above 1, logarithms yield positive results, acknowledging the exponential growth of numbers.

Understanding logarithms requires envisioning how exponential operations dominate numerical behavior, thus providing a versatile tool in fields ranging from science to engineering.