Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$g(x)=\log _{10} x$$

Step-by-Step Solution

Answer

The evaluations of the function at the given values are \(g(16) = 1.204\), \(g(1/4) = -0.602\), \(g(1/1000) = -3\) and \(g(100,000) = 5\).

Step 1: Evaluate \(g(16)\)

We note that \(16=10^{1.204}\). Therefore, \(g(16)=\log_{10}(16) = \log_{10}(10^{1.204}) = 1.204\)

Step 2: Evaluate \(g(1/4)\)

Observe that \(1/4 = 10^{-0.602}\). Hence, \(g(1/4)=\log_{10}(1/4) = \log_{10}(10^{-0.602}) = -0.602\)

Step 3: Evaluate \(g(1/1000)\)

We find that \(1/1000 = 10^{-3}\). So, \(g(1/1000)=\log_{10}(1/1000) = \log_{10}(10^{-3}) = -3\)

Step 4: Evaluate \(g(100,000)\)

Finally, recognize that \(100,000 = 10^{5}\). As a result, \(g(100,000)=\log_{10}(100,000) = \log_{10}(10^{5}) = 5\)

Key Concepts

Evaluating Logarithms Without a Calculator

Mastering the skill of evaluating logarithms without the aid of a calculator is a valuable asset, particularly when working with simple bases such as 10. To achieve this, one must understand that a logarithm, in essence, answers the question: 'To what exponent must we raise the base to obtain a certain number?'

For example, when asked to find \( \log_{10}16 \) without a calculator, you must think in terms of powers of 10. While 16 is not a perfect power of 10, recognizing that \( 10^{1} = 10 \) and \( 10^{2} = 100 \) can guide you to estimate that \( \log_{10}16 \) is slightly more than 1 but less than 2. Similarly, \( \log_{10}\frac{1}{4} \) requires understanding negative exponents and seeing \( \frac{1}{4} \) as \( 10^{-0.602} \) demonstrates this relationship.

Practice with various numbers can sharpen one's ability to recognize these relationships quickly and estimate logarithmic values without digital tools.

Properties of Logarithms

Logarithmic Identity

One of the fundamental properties of logarithms is the logarithmic identity: \( \log_{b}(b^{x}) = x \). This property allows for simplification and is pivotal when evaluating logarithms as the exponent can be directly extracted.

Product, Quotient, and Power Properties

Additionally, key properties which include the product \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \), quotient \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \), and power rules \( \log_{b}(x^{n}) = n \cdot \log_{b}(x) \) are invaluable tools for breaking down complex expressions into manageable parts.

Logarithm Base 10

The base 10 logarithm, usually referred to as the common logarithm and denoted \( \log \), holds a special place because our number system is based on powers of 10. It represents how many times you must multiply 10 to obtain a certain number.

For instance, \( \log_{10}(1000) = 3 \) because \( 10 \times 10 \times 10 = 1000 \). This particular relationship forms the backbone of many scientific and engineering calculations, as it easily translates orders of magnitude into a linear form, which is easier to comprehend and manipulate.

Exponential and Logarithmic Relationships

Exponential and logarithmic functions are inversely related; the logarithm function is the inverse of the exponential function. This means that if \( y = b^{x} \), then \( x = \log_{b}(y) \). It represents two different perspectives of the same process - one of growth (exponential) and the other of decoding that growth (logarithm).

Understanding this inverse relationship is crucial in solving exponential equations and can be particularly helpful in fields such as compound interest calculations in finance and population dynamics in biology, where exponential growth is frequently encountered.