Exponential and logarithmic functions Knowledge-Based Study

Study notes

1. Understanding Exponential Functions

   An exponential function is a function of the form:

   $$f(x) = b^x$$

   where the base b is a constant such that $b > 0$ and $b \neq 1$, and the exponent x is any real number.

   Key Characteristics:

   • Domain: $(-\infty, \infty)$
   • Range: $(0, \infty)$
   • y-intercept: $(0, 1)$, because $b^0 = 1$ for any non-zero base.
   • Horizontal Asymptote: The line $y = 0$ (the x-axis).

   Behavior Based on Base Value:

   • Exponential Growth ($b > 1$): The function increases strictly across its domain. As $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to 0$.
   • Exponential Decay ($0 < b < 1$): The function decreases strictly across its domain. As $x \to \infty$, $f(x) \to 0$, and as $x \to -\infty$, $f(x) \to \infty$.

   The natural exponential function uses Euler's number $e \approx 2.71828$ as its base, written as $f(x) = e^x$.

   Logarithmic Functions as Inverses

   Because exponential functions are strictly monotonic (one-to-one), they possess inverse functions. The inverse of an exponential function is a logarithmic function.

   For $b > 0$ and $b \neq 1$:

   $$y = \log_b(x) \iff b^y = x$$

   In this definition, $\log_b(x)$ represents the exponent to which the base b must be raised to produce the value x.

   Key Characteristics:

   • Domain: $(0, \infty)$ (the argument of a logarithm must always be strictly positive).
   • Range: $(-\infty, \infty)$
   • x-intercept: $(1, 0)$, because $\log_b(1) = 0$.
   • Vertical Asymptote: The line $x = 0$ (the y-axis).

   Common Notations:

   • Common Logarithm: $\log_{10}(x)$ is conventionally written simply as $\log(x)$.
   • Natural Logarithm: $\log_e(x)$ is written as $\ln(x)$.

   Logarithmic Identities and Rules

   The operational properties of logarithms derive directly from the laws of exponents. For positive numbers u, v, and base $b > 0, b \neq 1$:

   • Product Rule: $\log_b(uv) = \log_b(u) + \log_b(v)$
   • Quotient Rule: $\log_b\left(\frac{u}{v}\right) = \log_b(u) - \log_b(v)$
   • Power Rule: $\log_b(u^k) = k \cdot \log_b(u)$ for any real number k
   • Inverse Identities: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$ (for $x > 0$)

   Change of Base Formula

   To evaluate or transform logarithms into a different positive base a (where $a \neq 1$):

   $$\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)}$$

   This identity enables conversion between arbitrary bases and allows computation using natural or common logarithms.

   Techniques for Solving Equations

   1. Exponential Equations

   When solving exponential equations:

   • Like Bases: If $b^u = b^v$, then $u = v$. For example, $2^{3x-1} = 16 \implies 2^{3x-1} = 2^4 \implies 3x - 1 = 4 \implies x = \frac{5}{3}$.
   • Unlike Bases: Apply the natural logarithm to both sides. For example, to solve $5^x = 12$:
     $\ln(5^x) = \ln(12) \implies x \ln(5) = \ln(12) \implies x = \frac{\ln(12)}{\ln(5)}$.

   2. Logarithmic Equations

   When solving logarithmic equations, condense terms using properties and convert into exponential form or equate arguments:

   • If $\log_b(u) = \log_b(v)$, then $u = v$.
   • If $\log_b(u) = c$, rewrite as $u = b^c$.

   Crucial Verification Step: Always substitute potential solutions back into the original arguments of logarithmic terms. Any solution yielding a non-positive argument ($u \leq 0$) is an extraneous solution and must be discarded.