Question

solve the logarithmic equation for x. if there are no answers, enter none. \\(\log_{9}(4) + \log_{9}(x) = 1\\)

Explanation:

Step1: Apply log addition rule

Using the property \(\log_b(a) + \log_b(c)=\log_b(ac)\), we get \(\log_9(4x)=1\).

Step2: Convert to exponential form

Recall that \(\log_b(y)=k\) implies \(y = b^k\). So, \(4x=9^1\).

Step3: Solve for x

Simplify \(9^1 = 9\), then \(4x = 9\), so \(x=\frac{9}{4}\). Check the domain: \(x>0\) and \(4x>0\) (since log arguments must be positive), \(\frac{9}{4}>0\) is true.

Answer:

\(\frac{9}{4}\)