Understanding the Logarithmic Expression: log??(4) * 2 * log??(5)

The expression given here, log??(4) * 2 * log??(5), requires a careful breakdown of the properties of logarithms to understand its value. In this article, we will explore the application of logarithmic properties across this expression, focusing particularly on the base-10 logarithm (log??).

Step-by-Step Solution

Let's break down the expression step by step to find its value. The initial expression is:

log??(4) * 2 * log??(5)

Step 1: Appy the Power Rule of Logarithms

The first term, log??(4), can be viewed as part of a larger rule in logarithms. The property m * log?(a) log?(a^m) (also written as m log(a) log(a^m)) allows us to rewrite 2 * log??(5) as:

2 * log??(5) log??(5^2) log??(25)

Step 2: Simplify the Expression

Now we have:

log??(4) * log??(25)

The next step is to recognize that we can further simplify this using the property log?(b) * log?(c) log?(b * c).

log??(4) * log??(25) log??(4 * 25) log??(100)

Step 3: Evaluate the Final Expression

Now we need to evaluate log??(100). Since we are working with base-10 logarithms, we know that:

log??(100) log??(10^2) 2 * log??(10) 2 * 1 2

Thus, the value of the given expression is 2.

Summary

Let's summarize the key steps and the final value:

log??(4) * 2 * log??(5) log??(4) * log??(25) log??(100) 2

This exercise highlights the importance of logarithmic properties, specifically the power rule and the product rule, in simplifying and evaluating expressions. Understanding these properties is crucial for various applications in mathematics, science, and engineering.

Key Takeaways

The power rule of logarithms simplifies expressions involving coefficients. The product rule of logarithms can be used to combine logarithms of products. Base-10 logarithms are straightforward to evaluate when the argument is a power of 10.

Related Keywords

Logarithmic expression, logarithm, base 10