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Exponential and Logarithmic Functions

Exponential Functions

An exponential function is a mathematical function of the form $f (x) = a \cdot b^{x}$ , where:

$a$ is a constant,
$b$ is the base of the exponential, and
$x$ is the exponent.

Properties of Exponential Functions:

Domain: The domain of an exponential function is all real numbers, $(- \infty, \infty)$ .
Range: The range is $(0, \infty)$ for $b > 1$ and $(- \infty, 0)$ for $0 < b < 1$ .
Codomain: The codomain is typically all real numbers, but the actual range depends on the function's specific form.
Growth and Decay: If $b > 1$ , the function represents exponential growth. If $0 < b < 1$ , it represents exponential decay.

Example:

Population Growth: If a population of bacteria doubles every hour, the population at time $t$ hours can be modeled by $P (t) = P_{0} \cdot 2^{t}$ , where $P_{0}$ is the initial population.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is of the form $f (x) = \log_{b} (x)$ , where:

$b$ is the base of the logarithm,
$x$ is the argument of the logarithm.

Properties of Logarithmic Functions:

Domain: The domain is $(0, \infty)$ .
Range: The range is all real numbers, $(- \infty, \infty)$ .
Codomain: The codomain is typically all real numbers.
Inverse Relationship: The logarithmic function $\log_{b} (x)$ is the inverse of the exponential function $b^{x}$ .

Example:

pH Levels: The pH level of a solution is calculated using the logarithm: $pH = - \log_{10} [H^{+}]$ , where $[H^{+}]$ is the concentration of hydrogen ions.

Inverse Functions

The inverse of an exponential function $y = b^{x}$ is a logarithmic function $x = \log_{b} (y)$ , and vice versa.

Example:

Solving for Time: If you have an exponential growth model $P (t) = P_{0} \cdot e^{k t}$ and you want to solve for $t$ , you would use the natural logarithm: $t = \frac{\ln (P / P_{0})}{k}$ .

Domain and Range

Exponential Functions:

Domain: All real numbers, $(- \infty, \infty)$ .
Range: $(0, \infty)$ for $b > 1$ and $(- \infty, 0)$ for $0 < b < 1$ .

Logarithmic Functions:

Domain: $(0, \infty)$ .
Range: All real numbers, $(- \infty, \infty)$ .

Codomain

The codomain of both exponential and logarithmic functions is typically all real numbers, but the actual range depends on the specific function.

Logarithmic Formulas

Product Rule: $\log_{b} (x y) = \log_{b} (x) + \log_{b} (y)$
Quotient Rule: $\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$
Power Rule: $\log_{b} (x^{y}) = y \cdot \log_{b} (x)$
Change of Base Rule: $\log_{b} (x) = \frac{\log_{k} (x)}{\log_{k} (b)}$

Special Explanation of Change of Base Rule:

The change of base rule allows you to convert a logarithm of any base $b$ to a logarithm of another base $k$ . This is particularly useful when you need to calculate logarithms on a calculator that only supports base 10 or base $e$ (natural logarithm).

Examples of Logarithmic Functions in Daily Life

Earthquake Magnitude: The Richter scale measures the magnitude of earthquakes using a logarithmic scale. An earthquake that measures 7 on the Richter scale is ten times more powerful than one that measures 6.
Sound Intensity: Decibels (dB) measure sound intensity logarithmically. A sound that is 10 dB louder is perceived as twice as loud.
Information Theory: Logarithms are used to measure information entropy, which quantifies the amount of uncertainty or information content.