Question

If the graph of a logarithmic function \(f(x)=\log _{a} x\) where \(a>0\) and \(a \neq 1,\) is increasing, then its base is larger than _____.

Short answer

1

Step-by-step solution

- Understanding the Logarithmic Function

The function given is a logarithmic function of the form \(f(x) = \log_a(x)\), where \(a > 0\) and \(a eq 1\). This means that 'a' is the base of the logarithm, and \(x\) is the argument.

- Properties of the Logarithmic Function

A logarithmic function \(f(x) = \log_a(x)\) increases if the base \(a\) of the logarithm is greater than 1.

- Stating the Condition

Given that the function is increasing, we know from the properties of logarithmic functions that the base \(a\) must be larger than 1.

Key concepts

logarithmic properties

Logarithmic functions, typically written as \(f(x) = \log_a(x)\), have unique properties that differentiate them from other types of functions. One core property is based on the value of the base, \(a\). If \(a > 1\), the function is increasing, meaning that as \(x\) gets larger, \(f(x)\) also gets larger. Conversely, if \(0 < a < 1\), the function is decreasing.

A logarithmic function also passes the vertical line test, indicating it is a one-to-one function, and thus, every \(x\) value corresponds to exactly one \(y\) value. Understanding these properties helps us determine the behavior and characteristics of the function, which is essential for solving problems involving logarithms.

function behavior

The behavior of logarithmic functions can vary significantly depending on the base, \(a\). For example:

• If \(a > 1\), \(f(x)\) rises slowly to the right. This indicates that as the input value increases, the output value also increases, but at a decreasing rate.
• If \(0 < a < 1\), \(f(x)\) falls gradually to the right. Here, as the input value increases, the output value decreases.

This understanding is crucial when graphing logarithmic functions or solving equations involving logarithms. Remember, the horizontal asymptote for \(f(x) = \log_a(x)\) is always the x-axis (\(y = 0\)), indicating that the function approaches zero but never actually reaches it.

base of logarithm

The base of a logarithm, \(a\), is a critical aspect of logarithmic functions. It determines key attributes of the function, for instance:

• When \(a > 1\), the function is increasing.
• When \(0 < a < 1\), the function is decreasing.

The most common bases are \(e\) (the natural logarithm), \(10\) (common logarithm), and \(2\) (binary logarithm). Each of these bases has specific applications:

• Natural logarithms, written \(\ln(x)\), are used extensively in calculus and natural phenomena modeling.
• Common logarithms, written \(\log(x)\), are used in scientific notation and when dealing with orders of magnitude.
• Binary logarithms, written \(\log_2(x)\), are significant in computer science and information theory.

Understanding the base and its implications helps in appropriately applying logarithmic functions across different fields.