Questions: Write the following sum as a single logarithm. Assume all variables are positive. log3(a)+log3(a+4)= The answer format in lowercase characters is: logbase (number) Spaces in the answer are optional.

Solution Steps

To combine the sum of two logarithms into a single logarithm, we can use the logarithmic property that states \(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\). In this case, we have \(\log_3(a) + \log_3(a+4)\), which can be combined into a single logarithm by multiplying the arguments of the logarithms.

We start with the expression: \[ \log_{3}(a) + \log_{3}(a + 4) \] Using the logarithmic property \(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\), we can combine the two logarithms: \[ \log_{3}(a) + \log_{3}(a + 4) = \log_{3}(a \cdot (a + 4)) \]

Next, we simplify the argument of the logarithm: \[ a \cdot (a + 4) = a^2 + 4a \] Thus, we can rewrite the expression as: \[ \log_{3}(a^2 + 4a) \]

The final expression for the sum of the logarithms as a single logarithm is: \[ \log_{3}(a^2 + 4a) \]

Final Answer