Problem 11 - Technical Mathematics with Calculus

Question

Convert to exponential form. $$\log _{3} x=57$$

Step-by-Step Solution

Answer

Exponential form: \(3^{57} = x\)

Step 1: Understand Logarithmic Form

The logarithmic form \( \log_{b} a = c \) states that base \( b \) to the power of \( c \) equals \( a \).

Step 2: Apply the Definition to Exponential Form

Using the definition from Step 1, rewrite the logarithm \( \log_{3} x = 57 \) to its equivalent exponential form: \( 3^{57} = x \) which is the required conversion.

Key Concepts

Logarithms

A logarithm is a way to represent exponentiation in a different form, particularly when dealing with large numbers or complex calculations. It asks the question, 'To what exponent must we raise a certain base to get a certain number?' For instance, the logarithmic equation \( \log_{b} a = c \) means that the base \( b \) raised to the power \( c \) equals \( a \.\)

In our exercise, the notation \( \log_{3} x = 57 \) is a logarithmic form where 3 is the base, \( x \) is the number we want to find, and 57 is the exponent to which base 3 is raised. Understanding logarithms is crucial for solving exponential equations, which are prevalent in fields like physics, finance, and computer science where growth and decay rates are studied.

To make logarithms more approachable, think of them as the opposite of exponentiation. Just as subtraction is the inverse of addition, logarithms are the inverse of exponents. This concept is key to switching between logarithmic and exponential forms effectively.

Exponential Equations

Exponential equations involve variables in the exponent position, exemplified by \( b^{x} = a \.\) They appear in numerous scientific areas, including population growth models and the calculation of interest in finance. In our example, \( 3^{x} = a \) is an exponential equation with the base of 3.

Converting between logarithmic and exponential forms is an essential skill for solving these equations since logarithmic notation can simplify the process of isolating the exponent, which is often the variable we want to determine. By transforming \( \log_{3} x = 57 \) into its exponential form, we have \( 3^{57} = x \.\) Such a step makes it clearer how large the value of \( x \) is, as we can see that it is a result of exponentiating the base 3 by 57.

Remember, when manipulating these equations, keep the rules of exponents in mind, such as the product rule, quotient rule, and power of a power rule. These principles help manage different operations within exponential equations.

Mathematical Notation

Mathematical notation serves as a concise language to express complex mathematical ideas. It includes symbols and numbers arranged in a specific and standardized manner to relay information without ambiguity. For instance, in logarithms, the notation \( \log_{b} a \) tells us immediately that we are dealing with a logarithm with base \( b \) and that we are interested in the power to which \( b \) must be raised to obtain \( a \.\)

Throughout history, this notation has evolved to enhance clarity and efficiency in communication among mathematicians. With the example \( \log_{3} x = 57 \) we are conveying a wealth of information in a very compact form that, once understood, is incredibly powerful for solving problems.

Understanding how to use and interpret mathematical notation allows students to engage with more complex concepts, share their work with others, and understand the work of others. It is the fundamental building block for advancing in mathematics, and gaining proficiency in it can significantly aid in the study of all mathematical disciplines.