Use the power property to rewrite log(3x^9).
The correct answer and explanation is:
To use the power property of logarithms to rewrite: log(3×9)\log(3x^9)
We apply the logarithmic properties, particularly the product property and the power property:
Correct Answer:
log(3×9)=log(3)+log(x9)=log(3)+9log(x)\log(3x^9) = \log(3) + \log(x^9) = \log(3) + 9\log(x)
300-Word Explanation:
To simplify log(3×9)\log(3x^9), we apply logarithmic rules. The expression involves a product inside the logarithm and an exponent on the variable. Two properties are essential here:
- Product Property:
log(ab)=log(a)+log(b)\log(ab) = \log(a) + \log(b)
This means the log of a product is the sum of the logs.
- Power Property:
log(an)=nlog(a)\log(a^n) = n\log(a)
This means the exponent can be moved in front of the log as a multiplier.
Let’s apply the product property first to split the log of the product 3x93x^9: log(3×9)=log(3)+log(x9)\log(3x^9) = \log(3) + \log(x^9)
Next, apply the power property to the term log(x9)\log(x^9): log(x9)=9log(x)\log(x^9) = 9\log(x)
So, the entire expression becomes: log(3×9)=log(3)+9log(x)\log(3x^9) = \log(3) + 9\log(x)
This is the fully simplified form using logarithmic properties.
Why do we do this? Expressing logarithmic expressions in expanded form is helpful in many math and science contexts, such as solving equations, analyzing growth models, or simplifying calculus problems. It allows us to isolate terms and makes computation easier, especially when using a calculator or comparing values.
In summary, using the product property allows us to break apart multiplication inside a logarithm, and the power property allows us to pull exponents out in front. Together, they simplify complex logarithmic expressions like log(3×9)\log(3x^9) into more manageable parts: log(3)+9log(x)\boxed{\log(3) + 9\log(x)}