Problem 25 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{49} 343=\frac{3}{2} $$

Step-by-Step Solution

Answer

The exponential form is \( 49^{\frac{3}{2}} = 343 \).

Step 1: Understanding the Logarithmic Equation

The equation given is \( \log_{49} 343 = \frac{3}{2} \). This defines a relationship between a base, a logarithm result, and a power. Here, \( 49 \) is the base, \( 343 \) is the result of applying the logarithm, and \( \frac{3}{2} \) is the exponent or power to which the base needs to be raised to obtain the result.

Step 2: Translating Logarithm to Exponential Form

The logarithmic equation \( \log_{49} 343 = \frac{3}{2} \) can be translated into its equivalent exponential form by understanding that the base of the logarithm is the same as the base of the exponent. Therefore, \( 49 \) raised to the power of \( \frac{3}{2} \) equals \( 343 \). This gives the exponential equation: \( 49^{\frac{3}{2}} = 343 \).

Key Concepts

Understanding Logarithmic Equations

Logarithmic equations are a way to express the exponent that a specified base must be raised to, to obtain a certain number. Essentially, a logarithm is the inverse operation of exponentiation. In the exercise, we have the logarithmic equation \( \log_{49} 343 = \frac{3}{2} \). This tells us that the base \( 49 \), when raised to some power, results in \( 343 \). The logarithm, in this context, indicates what this power should be, which is \( \frac{3}{2} \).

• The number \( 49 \) is the base of the log.
• \( 343 \) is the number we get when the base is raised to the indicated power.
• The exponent, which is the result of the log operation, is \( \frac{3}{2} \).

By understanding and identifying these components, we can move forward with transforming this logarithmic statement into its exponential counterpart.

Conversion from Logarithmic to Exponential

Converting a logarithmic equation into an exponential form is a key skill in understanding the relationship between these two expressions. The process involves shifting from a log equation to a statement that expresses power.For the given equation \( \log_{49} 343 = \frac{3}{2} \), conversion to exponential form is straightforward:

• Identify the base of the logarithm, which in our case is \( 49 \).
• Recognize the exponent as the right side of the equation, \( \frac{3}{2} \).
• Translate it into the exponential expression: \( 49^{\frac{3}{2}} = 343 \).

Thus, the base \( 49 \), when raised to the power of \( \frac{3}{2} \), will equal \( 343 \). This is the same value in a different form, maintaining equivalence between logarithmic and exponential expressions.

Exploring the Role of Exponents

Exponents play a crucial role in both exponential and logarithmic equations. They tell us how many times the base is used in a multiplication. In the logarithmic equation \( \log_{49} 343 = \frac{3}{2} \), the exponent \( \frac{3}{2} \) is pivotal to converting it to exponential form.

• An exponent like \( \frac{3}{2} \) indicates a fractional power. It can be interpreted as a combination of both a square root and raising the number to a certain power.
• Expressing this, 49 raised to the power of \( \frac{3}{2} \) means \( \sqrt{49^3} \). It's useful to break down these operations:
  • First, calculate \( 49^3 \) (which is an intermediate step).
  • Then, find the square root of the result.
• This helps in further understanding and operating with exponential values, converting them from their logarithmic descriptions.

Exponents are fundamental in expressing large numerical values efficiently and compactly, emphasizing their importance in mathematical operations and conversions.