When working with logarithms, it is essential to understand how they can be simplified and expressed in different forms. In this article, we will explore the expression “log5 X 4 2” and determine which equivalent expression represents the same logarithmic value.
Logarithmic Properties
Before delving into the specific expression “log5 X 4 2,” it is crucial to review some fundamental logarithmic properties that will help us simplify and manipulate logarithmic expressions.
- Product Rule: $\log_b (xy) = \log_b x + \log_b y$
- Quotient Rule: $\log_b \left(\frac{x}{y}\right) = \log_b x – \log_b y$
- Power Rule: $\log_b (x^n) = n\log_b x$
- Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$
Understanding the Given Expression
The expression “log5 X 4 2” can be interpreted as $\log_5 (4^2)$. This means we are taking the base 5 logarithm of the value obtained by raising 4 to the power of 2.
Simplifying the Expression
To simplify the expression $\log_5 (4^2)$, we can use the power rule of logarithms which states that $\log_b (x^n) = n\log_b x$. Applying this rule to our expression, we get:
$$
\log_5 (4^2) = 2\log_5 4
$$
Equivalent Expression
Now, we need to determine which expression is equivalent to $2\log_5 4$. By inspection, we can see that it is not possible to simplify this expression further using the logarithmic properties mentioned earlier. Therefore, the equivalent expression to $\log_5 X 4 2$ is $2\log_5 4$.
Conclusion
In conclusion, the expression “log5 X 4 2” is equivalent to $2\log_5 4$ after simplification. Understanding logarithmic properties and how to manipulate logarithmic expressions is crucial in mathematics and various fields of study. Remember to practice working with logarithmic expressions to improve your skills and problem-solving abilities.
