Logarithmic Expressions: Simplifying Complex Equations
In this article, we will explore how to simplify complex logarithmic expressions, specifically the equation:
2 log (1/243) x 5 log (1/125) x 3 log (1/16)
To simplify this equation, we will apply the properties of logarithms and exponential functions.
Step 1: Rewrite the Equation using Logarithmic Properties
Using the property of logarithms that states log (a^b) = b log a, we can rewrite each term in the equation as:
2 log (1/243) = 2 log (1/3^5) = -10 log 3
5 log (1/125) = 5 log (1/5^3) = -15 log 5
3 log (1/16) = 3 log (1/4^2) = -6 log 4
Step 2: Simplify the Equation
Now, substitute the rewritten terms back into the original equation:
(-10 log 3) x (-15 log 5) x (-6 log 4)
To simplify further, we can use the property of logarithms that states log a x log b = log (a^b). Applying this property, we get:
-90 log (3^5) x -6 log 4
= -90 log 243 x -6 log 4
= -540 log (243 x 4)
= -540 log 972
Final Answer
Thus, the simplified equation is:
-540 log 972
By applying the properties of logarithms and exponential functions, we were able to simplify the complex logarithmic expression 2 log (1/243) x 5 log (1/125) x 3 log (1/16) to a single logarithmic term.
