Wagner Bernard
02/17/2023 · Senior High School
Write the following sum as a single logarithm. Assume all variables are positive. \( \log _{3}(x)+\log _{3}(x+6)= \) Preview The answer format in lowercase characters is: \( \log _{-} \)base (number) Spaces in the answer are optional.
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Step-by-step Solution
To combine the logarithms \( \log_{3}(x) + \log_{3}(x+6) \) into a single logarithm, we can use the property of logarithms that states:
\[
\log_{b}(a) + \log_{b}(c) = \log_{b}(a \cdot c)
\]
Applying this property, we have:
\[
\log_{3}(x) + \log_{3}(x+6) = \log_{3}(x \cdot (x+6))
\]
Now, we can simplify the expression inside the logarithm:
\[
x \cdot (x + 6) = x^2 + 6x
\]
Thus, we can write the sum as:
\[
\log_{3}(x^2 + 6x)
\]
In the required format, the answer is:
\[
\log_{3}(x^2 + 6x)
\]
So, the final answer in the specified format is:
\[
\log_{3}(x^2+6x)
\]
Quick Answer
\[
\log_{3}(x^2+6x)
\]
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