Problem 17 - A Graphical Approach to Precalculus with Limits

Question

March the logarithm in Column I with its value in Column II. Remember that \(\log _{a} x\) is the exponent to which a must be raised in onder to obtain \(x\). \(\mathbf{I}\) (a) \(\log _{2} 16\) (b) \(\log _{3} 1\) (c) \(\log _{10} 0.1\) (d) \(\log _{2} \sqrt{2}\) (e) \(\log _{e} \frac{1}{e^{2}}\) (f) \(\log _{1 / 2} 8\) \(\mathbf{II}\) A. 0 B. \(\frac{1}{2}\) C. 4 D. \(-3\) E. \(-1\) J. \(-2\)

Step-by-Step Solution

Answer

(a) C, (b) A, (c) E, (d) B, (e) J, (f) D.

Step 1: Understanding Logarithms

A logarithm \(\log_{a} x\) tells us the power (exponent) to which the base \(a\) must be raised to yield \(x\). For instance, \(\log_{2} 16\) asks what power 2 must be raised to get 16.

Step 2: Solve (a)

- \(\log_{2} 16 \): Since \( 2^4 = 16 \), the logarithm is 4. Therefore, \(\log_{2} 16 = 4\), which matches to C.

Step 3: Solve (b)

- \(\log_{3} 1 \): Any number raised to the power of 0 is 1 (since \(a^0 = 1\) for any \(a\), thus \(\log_{3} 1 = 0\). This matches to A.

Step 4: Solve (c)

- \(\log_{10} 0.1 \): \(0.1\) is equal to \(10^{-1}\), so \(\log_{10} 0.1 = -1\). This matches to E.

Step 5: Solve (d)

- \(\log_{2} \sqrt{2} \): \(\sqrt{2}\) is \(2^{1/2}\), so \(\log_{2} \(\sqrt{2}\) = 1/2\), which matches to B.

Step 6: Solve (e)

- \(\log_{e} \frac{1}{e^{2}} \): \( \frac{1}{e^{2}}\) is equal to \(e^{-2}\), so \(\log_{e} \frac{1}{e^{2}} = -2\). This matches to J.

Step 7: Solve (f)

- \(\log_{1/2} 8 \): This asks which power (exponent) of \(\frac{1}{2}\) equals 8. Since \(\left(\frac{1}{2}\right)^{-3} = 8\), \(\log_{1/2} 8 = -3\), which matches to D.

Key Concepts

Exponents

Exponents are a fundamental mathematical concept used to represent repeated multiplication. When you see an expression like "\(a^b\)," the number \(a\) is known as the base, while \(b\) is the exponent. This means that \(a\) is multiplied by itself \(b\) times. For example, \(2^4\) is equal to \(16\) because it represents \(2 \times 2 \times 2 \times 2\). In the context of logarithms, understanding how exponents work is crucial because a logarithm is essentially the inverse of exponentiation.

• An exponent that is zero (i.e., \(a^0\)) always equals 1, provided \(a\) is not zero.
• Negative exponents imply a reciprocal. For example, \(a^{-b} = \frac{1}{a^b}\).

Exponents are widely used across different areas of mathematics and can stretch to real numbers and even imaginary numbers in advanced mathematics.

Base

In mathematics, a base is the number that gets raised to a power by an exponent. The base is a central part of both exponential expressions and logarithms. It determines the number you are repeatedly multiplying when dealing with powers. For instance, in the expression \(8 = 2^3\), the base is 2, raised to the third power.

• The base of a logarithm, such as in \(\log_{10} x\), is the number \(10\) in this case.
• Changing the base of logarithms allows us to express the logarithms in various forms, such as converting \(\log_2\) to \(\log_{10}\), using the formula \(\log_a b = \frac{\log_c b}{\log_c a}\).

The concept of base is not only limited to logarithms but also extends into numerous mathematical fields like binary numbers, where the base is 2, and hexadecimal numbers, where the base is 16.

Logarithmic Functions

Logarithmic functions transform multiplication into addition, making them a handy tool in mathematical calculations. A logarithmic function is expressed as \(y = \log_b x\), where \(b\) is the base and \(x\) is the input value for which you want the logarithm. Logarithmic functions are the inverses of exponential functions, which means they "undo" exponentiation. If \(b^y = x\), then \(y = \log_b x\).

• They are perfect for understanding exponential growth and decay, commonly seen in finance for compound interest and in nature for population growth.
• The bases typically used are 10 (common logarithm), \(e\) (natural logarithm), and 2 for binary systems.

Overall, logarithmic functions retain their importance in fields such as engineering, astronomy, and computer science due to their ability to simplify complex mathematical models.

Logarithmic Identities

Logarithmic identities are essential rules that simplify the manipulation of logarithmic expressions. These identities help in solving equations and translating between logarithmic forms. Understanding these identities allows one to convert complex expressions into simpler equivalents.

• Product Identity: \(\log_b (xy) = \log_b x + \log_b y\)
• Quotient Identity: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Power Identity: \(\log_b (x^y) = y \cdot \log_b x\)
• Base Change Formula: \(\log_a b = \frac{\log_c b}{\log_c a}\)

With these identities, calculations involving logarithms become more feasible, especially when dealing with high powers or carrying out estimations in scientific calculations.