Problem 61 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{2} 8192=13 $$

Step-by-Step Solution

Answer

The exponential form of the given equation is \(2^{13} = 8192\).

Step 1: Identify the base, exponent and result

In the given equation \(\log_{2} 8192=13\), the base (b) is 2, the exponent (c) is 13 and the result (a) is 8192.

Step 2: Write in exponential form

Applying the conversion relationship, the exponential form of the given equation is \(2^{13} = 8192\).

Key Concepts

Logarithmic to Exponential Conversion

Converting equations from logarithmic to exponential form is an essential skill in mathematics. It allows for easier manipulation of expressions and deeper understanding of the relationships between numbers. When you see an equation like \( \log_b a = c \), it can be transformed into its exponential form. The conversion leads to the equation \( b^c = a \). Here, \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.

This relationship stems from the core concept of logarithms being the inverse functions of exponents. It can sometimes be tricky to remember at first, but once you understand the translation, it becomes straightforward to convert. Practicing this conversion by rewriting equations in exponential form is one of the best ways to strengthen your skill.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. For instance, \( 2^{13} \) means that the number 2 is multiplied by itself 13 times. Exponents are intimately related to logarithms because they serve as the inverse operations of each other.

Understanding exponents involves several key points:

• The base (in \( 2^{13} \), it is 2) shows which number is being multiplied.
• The exponent (13 in this case) tells how many times the base is used in the multiplication.
• The value of an expression with an exponent (like 8192) is the result of the multiplication.

Mastery of exponents requires practice, and converting between exponential and logarithmic forms is a good exercise in this area.

Logarithms

Logarithms are a fundamental tool in mathematics and are often used to simplify complex calculations. They essentially answer the question: "To what exponent must a certain base be raised, to yield a specific number?" Knowing the logarithm of a number can make expressing and solving equations more straightforward.

In the equation \( \log_2 8192 = 13 \), the logarithm informs us that 2 raised to the power of 13 equals 8192. Understanding logarithms involves recognizing their characteristics:

• They are the inverse operations of exponents, meaning they "undo" exponentiation.
• They are useful in finding unknown exponents when solving equations.
• They help in dealing with very large or very small numbers by reducing them to their exponents.

Logarithms can seem complex at first, but with practice, their utility and elegance can be easily appreciated.

Mathematical Equations

Mathematical equations are statements that express equality between two expressions. In dealing with equations involving logarithms and exponents, understanding both forms of the equation is key.

For example, the original equation \( \log_2 8192 = 13 \) and its exponential form \( 2^{13} = 8192 \) both express the same relationship in different ways. Here are some notable points about equations:

• Each form provides a different perspective and can offer insights into solving mathematical problems.
• The equality in an equation means that you can perform the same algebraic operation on both sides, assisting in finding solutions.
• Understanding various forms, such as exponential and logarithmic, is valuable for developing problem-solving strategies.

Equations in math are fundamental for expressing relationships and changes – mastering them can open doors to more advanced topics.