Simplifying Logarithmic Expressions: 1/3log6 + 1/2log6
In this article, we will explore how to simplify the logarithmic expression 1/3log6 + 1/2log6.
Properties of Logarithms
Before we dive into simplifying the expression, let's review some important properties of logarithms:
- Product Rule: loga(M) + loga(N) = loga(MN)
- Quotient Rule: loga(M) - loga(N) = loga(M/N)
- Power Rule: loga(M^p) = p * loga(M)
Simplifying 1/3log6 + 1/2log6
Now, let's simplify the given expression:
1/3log6 + 1/2log6
We can start by combining the two logarithmic terms using the Sum Rule, which states that loga(M) + loga(N) = loga(MN):
= log6^(1/3) + log6^(1/2)
Next, we can apply the Power Rule to simplify each term:
= log6^(1/3) + log6^(1/2) = (1/3)log6 + (1/2)log6
Combining Like Terms
We can combine the like terms by adding the coefficients:
= (1/3 + 1/2)log6
To add these fractions, we need a common denominator, which is 6. So, we can rewrite the expression as:
= (2/6 + 3/6)log6 = (5/6)log6
And that's the simplified expression!
Conclusion
In this article, we have successfully simplified the logarithmic expression 1/3log6 + 1/2log6 using the properties of logarithms. The final simplified expression is (5/6)log6.
