Problem 8 - Precalculus: A Concise Course

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25\). $$\log _{7} 343=3$$

Step-by-Step Solution

Answer

\(7^{3} = 343\)

Step 1: Identify the elements of the logarithmic equation

Here, we have a logarithm with base 7. The result of the logarithm is 343 and the value at the right-hand side of the equation is 3. These three elements will be used in the exponential equation.

Step 2: Convert the logarithmic equation to its exponential form

Substitute the base, result, and the right-hand side value into the exponential form. Therefore, the base 7 raised to the power of the right-hand side value 3 equals to the result 343.

Step 3: Write out the exponential form

Putting these numbers into the format gives us \(7^{3} = 343\).

Key Concepts

Exponential Form

The exponential form of a mathematical expression showcases the relationship between numbers, where a number, known as the base, is raised to a certain power or exponent. This is a way of expressing repeated multiplication. For example, if we have a base of 7 and an exponent of 3, the exponential form would be written as \(7^3\). This notation indicates that 7 is multiplied by itself three times: \(7 \times 7 \times 7\).
Exponential form is a compact way to express how many times a number should be multiplied by itself. It is frequently used in scientific notation and when dealing with large calculations. This form is particularly useful for handling problems involving growth rates, logarithms, and exponential decay.

Base and Exponent

In mathematics, the base and exponent are crucial components of exponential expressions. The base is the number that is multiplied by itself, and the exponent indicates how many times it appears in the multiplication.

• Base: The base is the main number that is affected by the power. In the example \(7^3\), the base is 7.
• Exponent: The exponent (sometimes called the power) tells you how many times to use the base in a multiplication. For \(7^3\), the exponent is 3, indicating that 7 is used three times as a factor in the multiplication process.

Understanding the roles of the base and exponent is important when manipulating exponential expressions, particularly when converting between different forms, such as logarithmic and exponential forms.

Conversion Between Forms

Converting between logarithmic and exponential forms is a helpful skill in algebra. A logarithm is essentially the inverse of taking a power. It tells you the power to which a certain base must be raised to produce a given number. For instance, if \(\log_7 343 = 3\), this is saying that 7 must be raised to the power of 3 to result in 343.
The conversion between logarithmic and exponential forms follows a basic principle:

• The logarithm \(\log_b a = c\) can be rewritten in exponential form as \(b^c = a\).

To convert a logarithmic expression to an exponential one, identify the base, the exponent (result of the logarithm), and the outcome (the number for which you've taken the log). In our example, converting \(\log_7 343 = 3\) to exponential form results in \(7^3 = 343\).
Grasping this conversion is crucial since many mathematical problems require rewriting expressions in different forms to simplify or solve equations effectively.