Simplifying Logarithmic Expressions: 1/3 log 27 - 2 log 1/3
In this article, we will explore how to simplify the logarithmic expression 1/3 log 27 - 2 log 1/3.
What is a Logarithm?
Before we dive into simplifying the expression, let's quickly review what a logarithm is. A logarithm is the inverse operation of exponentiation. It is defined as the power to which a base number must be raised to produce a given value. In other words, if x is the logarithm of y with base b, then b raised to the power of x is equal to y.
Simplifying the Expression
Now, let's simplify the given expression:
1/3 log 27 - 2 log 1/3
To simplify this expression, we need to use the following logarithmic properties:
- Power Rule: logₐ(xⁿ) = n logₐ(x)
- Quotient Rule: logₐ(x/y) = logₐ(x) - logₐ(y)
Using the power rule, we can rewrite the first term as:
1/3 log 27 = log 27^(1/3)
Since 27 = 3³, we can simplify the expression further:
log 27^(1/3) = log 3
Now, let's simplify the second term using the quotient rule:
2 log 1/3 = 2 log (1/3) = -2 log 3
Now, we can combine the two simplified terms:
log 3 - 2 log 3
By combining like terms, we get:
- log 3
Therefore, the simplified form of the expression 1/3 log 27 - 2 log 1/3 is - log 3.
Conclusion
In this article, we learned how to simplify the logarithmic expression 1/3 log 27 - 2 log 1/3 using the power rule and quotient rule of logarithms. We simplified the expression to its simplest form, which is - log 3.
