Introduction to Logarithmic Functions
Introduction
Logarithmic functions, or logarithms for short, are essential mathematical tools that arise from the study of exponential functions. They provide a way to solve equations involving exponential relationships, convert between different bases, and analyze the behavior of various phenomena. In this lesson, we will explore the basics of logarithmic functions, their properties, and their applications.
What is a Logarithmic Function?
A logarithmic function is the inverse of an exponential function. It represents the relationship between a given base and its exponent. The general form of a logarithmic function is written as $f(x) = log(base, x)$, where 'base' is a positive number greater than 1, and 'x' is the input value.
Application
Let us assume that you are investing some capital on an account that provides a $5\%$ interest rate compounded annually If you wanted to find out when would your money achieve a certain percentage growth you would use a logarithmic function
$Y(p) = \log(1+5% ,p)$ where $p$ is the percentage growth that you are looking for (e.g %200 when wanting to figure out when your money would double).
Properties of Logarithmic Functions:
Domain and Range
The domain of a logarithmic function is the set of all positive real numbers. The range depends on the base and is typically the set of all real numbers.
Inverse of Exponential Functions
Logarithmic functions and exponential functions are inverses of each other. If $y = a^x$, then $x = log(base, y)$, where the base s the same in both the exponential and logarithmic functions.
Logarithmic Identities:
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$log(base, 1) = 0$ The logarithm of 1 to any base is always 0.
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$log(base, base) = 1$ The logarithm of the base to the same base is always 1.
Logarithmic Laws
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Product Rule: $log(base, x * y) = log(base, x) + log(base, y)$
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Quotient Rule: $log(base, x / y) = log(base, x) - log(base, y)$
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Power Rule: $log(base, x^a) = a * log(base, x)$
Fields of application
Logarithmic functions find applications in various fields. Some common applications include:
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Computing pH levels in chemistry
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Measuring sound intensity with decibels
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Evaluating earthquake intensity using the Richter scale
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Analyzing population growth and decay models
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Studying the behavior of radioactive decay
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Solving exponential equations
Logarithmic functions provide a powerful tool for solving exponential equations, understanding exponential relationships, and analyzing various phenomena. As you delve deeper into logarithmic functions, you will encounter more advanced concepts and applications. Remember to practice and explore further to strengthen your understanding.