Indices, Exponentials, and Logarithms

• Indices, exponentials, and logarithms are fundamental concepts in mathematics. They are interconnected and used extensively in various fields like science, engineering, and finance.

Indices (Exponents)

• An index (or exponent, or power) indicates how many times a base number is multiplied by itself. For example, in (a^n), (a) is the base and (n) is the index.
• (a^n = a \times a \times a \times ...) ((n) times).

Basic Index Laws

• Product of powers: When multiplying powers with the same base, add the exponents: (a^m \times a^n = a^{m+n}).
• Quotient of powers: When dividing powers with the same base, subtract the exponents: (a^m \div a^n = a^{m-n}).
• Power of a power: When raising a power to another power, multiply the exponents: ((a^m)^n = a^{mn}).
• Power of a product: The power of a product is the product of the powers: ((ab)^n = a^n b^n).
• Power of a quotient: The power of a quotient is the quotient of the powers: ((a/b)^n = a^n / b^n).
• Zero exponent: Any non-zero number raised to the power of 0 is 1: (a^0 = 1) (where (a \neq 0)).
• Negative exponent: A number raised to a negative exponent is the reciprocal of the number raised to the positive exponent: (a^{-n} = 1/a^n).
• Fractional exponent: (a^{1/n}) is the (n)th root of (a).
• (a^{m/n}) is the (n)th root of (a^m), or equivalently, ((a^{1/n})^m).

Examples of Indices

• (2^3 = 2 \times 2 \times 2 = 8).
• (5^0 = 1).
• (3^{-2} = 1/3^2 = 1/9).
• (4^{1/2} = \sqrt{4} = 2).
• (8^{2/3} = (8^{1/3})^2 = (2)^2 = 4).

Exponential Functions

• An exponential function has the general form (f(x) = a^x), where (a) is a constant base and (x) is the variable exponent.
• The base (a) is typically a positive number not equal to 1 (i.e., (a > 0) and (a \neq 1)).

Key Properties of Exponential Functions

• Domain: The domain of an exponential function is all real numbers.
• Range: The range of an exponential function (f(x) = a^x) is all positive real numbers if (a > 0, a \neq 1).
• (y)-intercept: The graph of (f(x) = a^x) passes through the point (0, 1) because (a^0 = 1).
• Asymptote: If (a > 1), the x-axis (y = 0) is a horizontal asymptote as (x) approaches negative infinity. If (0 < a < 1), the x-axis (y = 0) is a horizontal asymptote as (x) approaches positive infinity.
• Growth/Decay: If (a > 1), the function represents exponential growth. If (0 < a < 1), the function represents exponential decay.

Graphs of Exponential Functions

• Exponential growth: When (a > 1), the graph increases rapidly as (x) increases.
• Exponential decay: When (0 < a < 1), the graph decreases rapidly as (x) increases.

Transformations of Exponential Functions

• Vertical shift: (f(x) = a^x + k) shifts the graph vertically by (k) units.
• Horizontal shift: (f(x) = a^{x - h}) shifts the graph horizontally by (h) units.
• Reflection: (f(x) = -a^x) reflects the graph across the x-axis. (f(x) = a^{-x}) reflects the graph across the y-axis.
• Vertical stretch/compression: (f(x) = c \cdot a^x) stretches the graph vertically if (c > 1) and compresses it if (0 < c < 1).

Logarithms

• A logarithm is the inverse operation to exponentiation. The logarithm of a number (x) to the base (a) is the exponent to which (a) must be raised to produce (x).
• Written as (log_a(x) = y), which means (a^y = x). (a) is the base of the logarithm, and (x) is the argument.

Common Logarithm Bases

• Base 10 (common logarithm): Denoted as (log_{10}(x)) or simply (log(x)).
• Base (e) (natural logarithm): Denoted as (log_e(x)) or (ln(x)), where (e) is approximately 2.71828.

Basic Logarithmic Properties

• Logarithm of 1: (log_a(1) = 0) because (a^0 = 1).
• Logarithm of the base: (log_a(a) = 1) because (a^1 = a).
• Product rule: (log_a(mn) = log_a(m) + log_a(n)).
• Quotient rule: (log_a(m/n) = log_a(m) - log_a(n)).
• Power rule: (log_a(m^k) = k \cdot log_a(m)).
• Change of base rule: (log_b(x) = log_a(x) / log_a(b)). This allows you to convert logarithms from one base to another.

Examples of Logarithms

• (log_{2}(8) = 3) because (2^3 = 8).
• (log_{10}(100) = 2) because (10^2 = 100).
• (ln(e) = 1) because (e^1 = e).

Logarithmic Functions

• A logarithmic function has the general form (f(x) = log_a(x)), where (a) is a positive constant not equal to 1.

Key Properties of Logarithmic Functions

• Domain: The domain of a logarithmic function is all positive real numbers.
• Range: The range of a logarithmic function is all real numbers.
• (x)-intercept: The graph of (f(x) = log_a(x)) passes through the point (1, 0) because (log_a(1) = 0).
• Asymptote: The y-axis (x = 0) is a vertical asymptote.
• Increasing/Decreasing: If (a > 1), the function is increasing. If (0 < a < 1), the function is decreasing.

Graphs of Logarithmic Functions

• When (a > 1), the graph increases slowly as (x) increases.
• When (0 < a < 1), the graph decreases as (x) increases, approaching the y-axis.

Transformations of Logarithmic Functions

• Vertical shift: (f(x) = log_a(x) + k) shifts the graph vertically by (k) units.
• Horizontal shift: (f(x) = log_a(x - h)) shifts the graph horizontally by (h) units.
• Reflection: (f(x) = -log_a(x)) reflects the graph across the x-axis. (f(x) = log_a(-x)) reflects the graph across the y-axis.
• Vertical stretch/compression: (f(x) = c \cdot log_a(x)) stretches the graph vertically if (c > 1) and compresses it if (0 < c < 1).

Solving Exponential Equations

• Use logarithms to solve exponential equations where the variable is in the exponent.
• If (a^x = b), then (x = log_a(b)). You can also take the logarithm of both sides of the equation to solve for (x).
• Example: Solve (2^x = 7). Taking the natural logarithm of both sides gives (ln(2^x) = ln(7)), so (x \cdot ln(2) = ln(7)), and (x = ln(7) / ln(2)).

Solving Logarithmic Equations

• Convert logarithmic equations to exponential form to solve for the variable.
• If (log_a(x) = y), then (x = a^y).
• Example: Solve (log_2(x) = 5). Then (x = 2^5 = 32).
• Be careful to check for extraneous solutions, especially when dealing with logarithms, as you cannot take the logarithm of a non-positive number.

Applications

• Exponential growth: Models population growth, compound interest, etc. Formula: (A = P(1 + r)^t), where (A) is the final amount, (P) is the principal, (r) is the rate, and (t) is the time.
• Exponential decay: Models radioactive decay, depreciation, etc. Formula: (A = A_0 e^{-kt}), where (A) is the final amount, (A_0) is the initial amount, (k) is the decay constant, and (t) is the time.
• Logarithms are used in scales like the Richter scale (earthquakes) and the pH scale (acidity).

Common Mistakes

• Incorrectly applying index laws (e.g., adding exponents when bases are different).
• Forgetting that (a^0 = 1).
• Not recognizing the domain restrictions of logarithmic functions (argument must be positive).
• Not checking for extraneous solutions in logarithmic equations.
• Confusing exponential growth and decay.