Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(f(x)=\log _{2} x\) and \(g(x)=\log x\) are examples of ____ functions.

Step-by-Step Solution

Answer

Logarithmic functions.

Step 1: Identifying Function Type

Both functions, \(f(x)=\log _{2} x\) and \(g(x)=\log x\), represent the mathematical operation of taking a logarithm. The specific feature of these functions is that they transform a multiplicative relationship into an additive one, which means they exhibit properties related to logarithmic transformations. Therefore, these are examples of logarithmic functions.

Step 2: Checking the Base

The base of the logarithm indicates how the logarithmic function grows. In \(f(x)=\log _{2} x\), the base is 2, making it a logarithmic function with base 2. In \(g(x)=\log x\), the base is 10, which is often the implied base when no base is specified, meaning it is a common logarithm, also a type of logarithmic function.

Step 3: Verifying by Definition

According to the mathematical definition, a logarithmic function is any function that can be written in the form \(\log_b x\), where \(b\) is the base and \(x\) is the argument of the logarithm. Since both \(f(x)\) and \(g(x)\) conform to this definition with specific bases, we confirm they are logarithmic functions.

Key Concepts

Logarithm

A logarithm answers the question: "To what exponent must we raise a given base to produce a certain number?" If you think of the question in terms of multiplication, a logarithm is the inverse operation of exponentiation. For instance, if you have the equation \(b^y = x\), then the logarithmic form is \(\log_b x = y\). Logarithms help us understand how numbers relate to each other on a multiplicative scale.
They are often used in various real-world applications, like calculating growth rates or solving problems in science and engineering.

• Logarithmic Function Notation: Typically written as \(\log_b x\), where \(b\) is the base, \(x\) is the argument, and the solution is the exponent needed to achieve \(x\).
• Additive Property: One key property is \(\log_b(mn) = \log_b m + \log_b n\). This property reflects how logarithms convert multiplication into addition.

The use of logarithms allows simplification of complex multiplication and division problems, turning them into easier addition and subtraction calculations.

Base of Logarithm

The base of a logarithm greatly affects the behavior and appearance of the logarithmic function. Different bases change the rate at which the function grows. The base is shown as the subscript in \(\log_b x\). Think of it as a fixed point that all calculations orbit around.
Using a base changes how dramatically the logarithmic value increases as the input numbers increase.

• Common Bases: Bases like 2, \(e\), and 10 are often used. Each has unique applications and properties.
• Effect on Graph: A smaller base means the logarithm grows slower, while a larger base means quicker growth.

In mathematics, changing the logarithmic base can provide deeper insights or simplify calculations, as seen when using base 2 in binary computations or base \(e\) in natural logarithms.

Common Logarithm

When you see a logarithm written simply as \(\log x\) without a base specified, it's assumed to be a common logarithm. Having base 10, it is the standard in many fields, typically connected to our decimal number system.
This makes it exceptionally useful for calculations that involve the conversion between scales, like those in scientific notation.

• Relation to Exponents: This log gives us the number of 10s to multiply to reach a number, which is why it's ubiquitous in scientific and financial contexts.
• Applications: Common logs are pivotal in measuring the strength of earthquakes (Richter scale) or sound intensity (decibels).

Common logarithms simplify many mathematical tasks by connecting easily with the human tendency to think in tens, improving accessibility in educational and practical applications.