Introduction to Logarithmic Properties and Equations

Logarithmic properties and equations are essential tools in mathematics, often used in various fields such as engineering, physics, and computer science. This article delves into the solution of equations that involve logarithms, specifically focusing on the expression 10 log 10.5. We will break down the process step-by-step to understand the underlying principles and properties of logarithms.

Solving 10 log 10.5

Let us begin by solving the expression 10 log 10.5. The expression can be solved using logarithmic properties. Here's the breakdown:

Step 1: Simplify the Numerator

10 log 10.5

First, we simplify the numerator. We can rewrite 10.5 as a fraction:

10 log 10.5 10 log (5/10)

Step 2: Apply Logarithmic Property

Using the logarithmic property that states log (a/b) log a - log b, we can further simplify:

10 log (5/10) 10 (log 5 - log 10)

Step 3: Simplify Log 10

We know that log 10 1, therefore:

10 (log 5 - log 10) 10 (log 5 - 1)

Final Expression

The expression simplifies to:

10 log 5 - 10

Solving 10log10 0.5

Let us now solve the expression 10log10 0.5. Here's the process:

Step 1: Direct Application of Logarithmic Identity

The expression 10log10 0.5 can be simplified directly using the logarithmic identity:

blogb n n

Step 2: Direct Substitution

Substituting the values:

10log10 0.5 0.5

Conclusion

By understanding and applying logarithmic properties, we can simplify and solve complex expressions involving logarithms. This knowledge is crucial for solving various mathematical and scientific problems. Whether you need to simplify 10 log 10.5 or solve expressions like 10log10 0.5, the principles of logarithms provide a clear and direct route to the solutions.

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Additional Resources

For further reading and exercises, consider exploring online resources such as Khan Academy, MIT OpenCourseWare, or any mathematical textbook on logarithms and logarithmic equations.