Finding the Minimum Value of a Logarithmic Function

The question asks for the minimum value of the function \(f(x) = \log_{10}(x^2 + 2x + 11)\). To find the minimum value of this logarithmic function, we first need to analyze its structure.

The function \(f(x)\) is a logarithmic function with base 10. The base 10 is greater than 1. A logarithmic function with a base greater than 1 is an increasing function. This means that as its argument increases, the value of the function also increases. Conversely, as its argument decreases, the value of the function also decreases.

Therefore, the minimum value of \(f(x) = \log_{10}(x^2 + 2x + 11)\) will occur when the argument of the logarithm, which is the quadratic expression \(g(x) = x^2 + 2x + 11\), is at its minimum value.

Minimizing the Quadratic Argument

Let's find the minimum value of the quadratic function \(g(x) = x^2 + 2x + 11\). This is a quadratic in the standard form \(ax^2 + bx + c\), where \(a = 1\), \(b = 2\), and \(c = 11\).

Since the coefficient of the \(x^2\) term, \(a = 1\), is positive (\(a > 0\)), the parabola represented by this quadratic opens upwards. The minimum value of an upward-opening parabola occurs at its vertex.

The x-coordinate of the vertex of a parabola \(ax^2 + bx + c\) is given by the formula \(x = \frac{-b}{2a}\).

Let's calculate the x-coordinate of the vertex for \(g(x) = x^2 + 2x + 11\):

\(x = \frac{-2}{2 \times 1} = \frac{-2}{2} = -1\)

So, the minimum value of the quadratic \(g(x)\) occurs at \(x = -1\).

Now, we find the minimum value of \(g(x)\) by substituting \(x = -1\) into the expression:

\(g(-1) = (-1)^2 + 2(-1) + 11\)

\(g(-1) = 1 - 2 + 11\)

\(g(-1) = -1 + 11\)

\(g(-1) = 10\)

Thus, the minimum value of the argument \(x^2 + 2x + 11\) is 10.

Calculating the Minimum Logarithmic Value

Now that we have the minimum value of the argument, we can find the minimum value of the function \(f(x)\) by substituting this value into the logarithm:

Minimum value of \(f(x) = \log_{10}(\text{minimum value of } (x^2 + 2x + 11))\)

Minimum value of \(f(x) = \log_{10}(10)\)

The logarithm \(\log_{10}(10)\) asks "To what power must 10 be raised to get 10?". The answer is 1, because \(10^1 = 10\).

\(\log_{10}(10) = 1\)

Therefore, the minimum value of the function \(f(x) = \log_{10}(x^2 + 2x + 11)\) is 1.

Summary of Steps

Here's a quick summary of how we found the minimum value:

1. Identified that the minimum of \(f(x) = \log_{10}(g(x))\) occurs when \(g(x)\) is minimum, because the base (10) is > 1.
2. Analyzed the argument \(g(x) = x^2 + 2x + 11\) as a quadratic function.
3. Found the x-coordinate of the vertex of the quadratic using \(x = -b/(2a)\).
4. Calculated the minimum value of the quadratic by substituting the vertex's x-coordinate back into \(g(x)\).
5. Substituted the minimum value of the quadratic into the logarithmic function \(f(x)\) to find its minimum value.

The minimum value of the function \(f(x) = \log_{10}(x^2 + 2x + 11)\) is 1.

Revision Table - Logarithm Minimum Value

Additional Information - Analyzing Logarithmic Functions

When analyzing logarithmic functions of the form \(f(x) = \log_b(g(x))\), understanding the base \(b\) and the argument \(g(x)\) is crucial.

• Base \(b\):
  • If \(b > 1\), the logarithm is an increasing function. \(f(x)\) increases as \(g(x)\) increases. Minimum \(f(x)\) occurs at minimum \(g(x)\), maximum \(f(x)\) occurs at maximum \(g(x)\).
  • If \(0 < b < 1\), the logarithm is a decreasing function. \(f(x)\) decreases as \(g(x)\) increases. Minimum \(f(x)\) occurs at maximum \(g(x)\), maximum \(f(x)\) occurs at minimum \(g(x)\).
• Argument \(g(x)\):
  • The argument of a logarithm must always be positive. So, \(g(x) > 0\). This defines the domain of the function \(f(x)\).
  • For \(g(x) = x^2 + 2x + 11\), we found the minimum value is 10. Since 10 is positive, the argument is always positive for all real x, meaning the domain of \(f(x)\) is all real numbers. We can verify this by checking the discriminant of the quadratic: \(\Delta = b^2 - 4ac = 2^2 - 4(1)(11) = 4 - 44 = -40\). Since \(\Delta < 0\) and \(a > 0\), the quadratic is always positive.
  • The minimum or maximum value of \(g(x)\) is often found using calculus (finding critical points by setting the derivative to zero) or, for quadratics, using properties of parabolas (vertex).

By combining the behavior of the base and the range of the argument, we can determine the range and minimum/maximum values of the overall logarithmic function.