Understanding Exponential and Logarithmic Functions

Exponential Meaning

In A-level mathematics, the term “exponential” refers to exponential functions or exponential growth/decay. Typically, the first chapter introducing pure mathematics begins with an explanation of exponential and logarithmic functions. The following 3 points are the key uses of exponentials: 

Exponential Functions: An exponential function is a mathematical function of the form f(x) = a^x, where “a” is a constant greater than 0 and not equal to 1, and “x” is the variable. The exponent, “x,” can be any real number. Exponential functions have several distinctive properties, including rapid growth or decay and the presence of a horizontal asymptote.

Exponential Growth: Exponential growth refers to a situation where a quantity increases rapidly over time. In the context of A-level mathematics, exponential growth typically involves the application of exponential functions to model scenarios such as population growth, compound interest, or the growth of a bacterial population.

Exponential Decay: Exponential decay refers to a situation where a quantity decreases rapidly over time. It is the opposite of exponential growth. Exponential decay is often used to model phenomena such as radioactive decay, the decrease in the value of an investment over time, or the decay of a medication in the body.

Exponential functions and exponential growth/decay are important concepts in mathematics and have applications in various fields, including finance, biology, physics, and economics. Understanding exponential functions allows for the analysis and prediction of quantities that exhibit exponential behavior, providing insights into real-world phenomena.

Logarithmic Meaning

In A-level mathematics, logarithmic refers to logarithmic functions and logarithms. Logarithms are mathematical operations that represent the exponent or power to which a fixed number, called the base, must be raised to obtain a given value. Logarithmic functions are the inverse of exponential functions.

The common logarithm, also known as the base-10 logarithm, is most commonly used in A-level mathematics. It uses the base 10 and is denoted as log(x) or simply ㏑(x).

Key concepts related to logarithmic functions include:

Logarithmic Identity: The logarithmic identity states that log(a × b) = log(a) + log(b). It means that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.

Logarithmic Properties: Logarithmic functions have several properties, including log(1) = 0, log(a^x) = xlog(a), and log(a/b) = log(a) – log(b). These properties are useful for simplifying and manipulating logarithmic expressions.

Solving Equations: Logarithms are often used to solve exponential equations. By taking logarithms on both sides of an exponential equation, it is possible to transform it into a simpler form and solve for the unknown variable.

Exponential and Logarithmic Forms 

For any positive number a, a ≠ 1. 

(Index Form) y = ex, which can be converted to  x = loge y, or x = ㏑ y. (Logarithmic Form) 

Laws of Logarithm 

Multiplication 

xy = x + y.

 

Division 

㏑ x/y = ㏑ x – y.

 

Power 

㏑ xn = nx. 

Math Questions  

Logarithmic: 

Solve the following equation, 

2㏑(X+2) = ㏑x+3㏑2

Solution:

㏑( x + 2)2 = ln x (23)

X2 + 4X + 4 = 8X 

(X – 2)2 = 0

X= 2

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