Solving Logarithmic Equations: How to Calculate log10 (1/4)
Logarithms are an essential tool in mathematics and are used to solve various real-world problems, from calculating the pH of a solution in chemistry to understanding sound levels in decibels. In this article, we will walk through the process of solving the logarithmic equation log10(1/4) using the properties of logarithms. Understanding this step-by-step approach will help you grasp the fundamental concepts of logarithms and improve your skills in solving similar equations.
Understanding Logarithms
Logarithms are the inverse of exponentiation. They answer the question: to which power must a base be raised to obtain a particular number? For example, in the equation log10(100) 2, it means that 10 must be raised to the power of 2 to get 100. This relationship can be expressed as:
102 100
Step-by-Step Solution for log10(1/4)
To solve the equation log10(1/4), we will use the properties of logarithms. Here is a detailed step-by-step method:
Step 1: Apply the Quotient Rule
The quotient rule of logarithms states that:
logb(a/c) logb(a) - logb(c)
Applying this rule to our equation:
log10(1/4) log10(1) - log10(4)
Step 2: Simplify log10(1)
Recall that any number raised to the power of 0 is 1. Therefore:
100 1
Using this, we can determine that:
log10(1) 0
Substituting this back into our equation:
log10(1/4) 0 - log10(4) -log10(4)
Step 3: Simplify log10(4)
Next, we need to calculate log10(4). We can express 4 as: 4 22. Using the power rule of logarithms, which states that:
logb(an) n logb(a)
We can rewrite log10(4) as:
log10(4) log10(22) 2 log10(2)
From a calculator or logarithm table, we find that:
log10(2) ≈ 0.3010
Therefore:
log10(4) ≈ 2 × 0.3010 0.6020
Step 4: Combine the Results
Substitute the calculated value back into the equation:
log10(1/4) ≈ -0.6020
Thus, the final result is:
log10(1/4) ≈ -0.6020
Conclusion
Understanding and applying the properties of logarithms can simplify complex mathematical problems. In this article, we have demonstrated the step-by-step process to solve the equation log10(1/4). By using the properties of logarithms, such as the quotient rule and the power rule, along with a basic understanding of logarithm values, we can efficiently and accurately solve such problems.