Questions: Condense the expression to a single logarithm. Write fractional exponents as radicals. Assume that all variables represent positive numbers. 2(log3(7) + log3(a)) - log3(4) =

Solution Steps

To condense the expression to a single logarithm, we can use the properties of logarithms. First, apply the product rule to combine the terms inside the parentheses. Then, use the power rule to handle the coefficient outside the parentheses. Finally, apply the quotient rule to combine the terms into a single logarithm.

We start with the expression: \[ 2\left(\log_{3}(7) + \log_{3}(a)\right) - \log_{3}(4) \] Using the properties of logarithms, we can expand this to: \[ 2\log_{3}(7) + 2\log_{3}(a) - \log_{3}(4) \]

Next, we apply the product rule and power rule of logarithms: \[ 2\log_{3}(7) + 2\log_{3}(a) = \log_{3}(7^2) + \log_{3}(a^2) = \log_{3}(49) + \log_{3}(a^2) \] Thus, we can combine these terms: \[ \log_{3}(49a^2) \]

Now, we incorporate the subtraction of \(\log_{3}(4)\): \[ \log_{3}(49a^2) - \log_{3}(4) = \log_{3}\left(\frac{49a^2}{4}\right) \]

Final Answer