Question

Rewrite the equation in exponential form. (See Example 1.) \(\log _7 343=3\)

Short answer

The exponential form of the equation \(\log _7 343 = 3\) is \(7^3 = 343\).

Step-by-step solution

Identify the base, result and power

First, identify the base, result and power in the logarithmic form. In \(\log _7 343 = 3\), 7 is the base, 343 is the result, and 3 is the power.

Convert to exponential form

Using the rule that transforms a logarithm into an exponent (\(b^c = a\)), rewrite the equation in exponential form. Thus, the equation becomes \(7^3 = 343\).

Key concepts

Logarithmic Functions

Logarithmic functions offer a way to solve equations where the exponent is unknown. In simpler terms, logarithms help you find the exponent to which a base number is raised to get a particular result. Let's break it down with an example like \(\log _7 343 = 3\).

• The logarithm tells us that 7 (the base) raised to some power gives us 343.
• Here, the answer, or logarithm, is 3. This means 7 raised to the power of 3 equals 343.

This functionality is essential in many fields like engineering, sciences, and even in calculating interest rates over time. It simplifies expressing very large or very small numbers, making it easier to visualize and work with these values. Every logarithm has a base, and in our example, that base is 7. It helps to remember that logarithms are essentially the inverse operation of exponentiation.

Base and Power

The concepts of base and power are central to understanding both exponents and logarithms. In any mathematical operation involving powers, two main components need to be identified:

• Base: This is the number that gets multiplied repeatedly.
• Power: This represents how many times the base is multiplied by itself.

In the context of our solved exercise from \(\log _7 343 = 3\), we identified that 7 is the base and 3 is the power. This is crucial, as it guides us to transform logarithmic expressions into exponential form.
The exponential equation equivalent of the original logarithmic form is structured as \(b^c = a\), where "b" is the base and "c" is the power, while "a" is the result. As seen, applying these components turns \(\log _7 343 = 3\) into \(7^3 = 343\). Understanding this conversion gives clarity on how numbers interact in these forms and shows the interchangeable nature of bases and powers between logarithmic and exponential functions.

Exponential Equations

Exponential equations involve expressions where variables appear as exponents. These equations are powerful tools in mathematics, enabling you to work with exponential growth or decay situations, be it population dynamics, radioactive decay, or financial growth. In our context, we looked at the conversion of a logarithmic statement into an exponential format.
The conversion process uses the basic form \(b^c = a\), showing an exponential equation. For example, our logarithmic equation \(\log _7 343 = 3\) transforms into \(7^3 = 343\): 7, raised to the power of 3, becomes 343. This shows that exponential forms are direct representations of how many times a base should be multiplied to reach a certain value. They offer a straightforward approach to visualize and solve problems involving exponential growth, compounding interest, or decay.

• Exponential form clarifies the relationship between numbers.
• It is particularly useful in simplifying calculations and solving equations.

Grasping how to transition from logarithmic to exponential forms unlocks many paths in understanding complex mathematical and real-world phenomena.