Difference Quotient If f(x)=loga x, show that (f(x+h)-f(x))/(h)=loga(1+(h)/(r))^1 / h, h ≠0

Name: Problem 104, Chapter 6: Algebra and Trigonometry
Uploaded: 2021-06-23T13:18:17-08:00
Duration: 1 min 8 s
Channel: Dharmendra Jain
Description: Difference Quotient If f(x)=loga x, show that (f(x+h)-f(x))/(h)=loga(1+(h)/(r))^1 / h, h ≠0

step 1 In this question we are going to understand the basic concept of linear speed and angular speed.

Difference Quotient If f(x)=loga x, show that...

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that represents the average rate of change of a function over an interval. It is the expression (f(x+h) - f(x))/h for h ? 0 and serves as the basis for defining the derivative, which is the limit of the difference quotient as h approaches zero.

Logarithm Function

A logarithm function, such as f(x) = log?(x), is the inverse of the exponential function. Its properties allow us to convert multiplicative relationships into additive ones, making it easier to work with exponential growth or decay. When working with such functions, understanding their behavior and transformation properties is crucial.

Properties of Logarithms

Logarithms have several key properties, including the fact that the difference of two logarithms with the same base (log?(b) - log?(c)) can be expressed as the logarithm of a quotient (log?(b/c)). Additionally, the logarithm of an exponentiated value, such as log?(b?), simplifies to c · log?(b). These properties are essential for manipulating expressions involving logarithmic functions.

Exponential and Limit Relationships

Algebraic manipulation involving exponents and limits plays a crucial role when dealing with expressions derived from the difference quotient, especially when re-expressing the quotient in logarithmic form. Recognizing patterns that connect small increments (h) and their limit behavior allows one to establish connections between the difference quotient and the derivative via logarithmic transformations.

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