Solving Logarithmic Equations: A Step-by-Step Guide
Understanding how to solve logarithmic equations is a fundamental skill in mathematics with numerous practical applications in science, engineering, and data analysis. This article will walk you through the detailed steps needed to solve the equation log8 m log8 (1/6) 2/3. By the end of this guide, you will have a clear understanding of the properties of logarithms and how to apply them to solve similar equations.
Step-by-Step Solution
To solve the equation log8 m log8 (1/6) 2/3, we will use the properties of logarithms to break down the problem into more manageable parts.
Step 1: Combine the Logarithms
First, we use the property of logarithms that states: loga b loga c loga (b cdot c) Applying this property to our equation, we get:
log8 m log8 (1/6) log8 (m cdot (1/6)) log8 (m/6)
So, the equation simplifies to:
log8 (m/6) 2/3
Step 2: Convert to Exponential Form
Next, we convert the logarithmic equation to its exponential form to eliminate the logarithm. The general form loga x y is equivalent to x ay.
Applying this conversion to our equation, we get:
m/6 82/3
Step 3: Calculate 82/3
To simplify 82/3, we can use the property that (am)n am*n. Since 8 can be expressed as 23, we have:
82/3 (23)2/3 23*(2/3) 22 4
Thus, the equation becomes:
m/6 4
Step 4: Solve for m
Multiplying both sides of the equation by 6, we find:
m 4 * 6 24
Conclusion
Therefore, the value of m is 24.
In summary, we solved the logarithmic equation log8 m log8 (1/6) 2/3 step by step using the properties of logarithms and converting the logarithmic form to an exponential form. The solution demonstrated the practical application of these mathematical principles in solving real-world problems.
Additional Examples
Example 1: Solve the equation log8 m log8 (1/6) 2/3.
Example 2: Solve the equation log3 (m/6) 2/3.