Converting Logarithmic Equations to Exponential Form and Solving for Variables

Understanding how to convert logarithmic equations into exponential form and solve for variables is a crucial skill in algebra. This guide will walk you through the process of converting logb2 1/2 into exponential form, and then solving for the variable b. Let's dive into the details.

Converting the Logarithmic Equation to Exponential Form

The given logarithmic equation is:

logb2 1/2

This equation tells us that the base b raised to the power of 1/2 equals 2. In other words:

b1/2 2

Here, the exponent 1/2 indicates a square root operation, and the base b raised to this exponent equals 2.

Step-by-Step Solution

Let's solve this step by step:

Start with the given logarithmic equation:

logb2 1/2

Rewrite it in exponential form:

b1/2 2

To eliminate the exponent, square both sides of the equation:

22 (b1/2)2

This simplifies to:

4 b

Therefore, the solution is:

b 4

Understanding the Exponential Form and Squaring Both Sides

Recall that the logarithmic equation logb2 1/2 can be interpreted as:

bx 2, where x 1/2. In exponential form, this is:

b1/2 2

To solve for b, we can square both sides of the equation to eliminate the exponent:

22 (b1/2)2

This simplifies to:

4 b

Additional Examples and Practice

To reinforce your understanding, consider solving a similar problem:

Convert and solve the equation logb3 2 into exponential form and find the value of b.

Steps:

Solution:

b2 3

Take the square root of both sides:

b 31/2 or b sqrt{3}

Conclusion

Mastering the conversion from logarithmic form to exponential form and solving for variables is a foundational skill in algebra. By following a systematic approach and practicing similar problems, you can become proficient in these types of equations. Whether you're working on homework problems or preparing for an exam, this skill will be invaluable.

Keywords: logarithmic equation, exponential form, solving for variable