Questions: 6.6 HW Log Exponential Equations Question 12, 6.6.39 HW Score: 38.1 % Part 1 of 2 Points: 0 of 1 Solve the following logarithmic equation. Express irrational solutions in the exact form and as decimals. 2 log4(x+1)=5 log4 3+log4 9 The solution set is . (Use a comma to separate answers as needed. Simplify your answers. Type exact answers, using radicals as needed.)

Solution Steps

To solve the given logarithmic equation, we can use the properties of logarithms to simplify and solve for \( x \). First, we will combine the logarithms on the right side of the equation using the property \( a \log_b c = \log_b c^a \). Then, we will equate the arguments of the logarithms since the bases are the same, and solve the resulting equation for \( x \).

The given equation is: \[ 2 \log_{4}(x+1) = 5 \log_{4} 3 + \log_{4} 9 \] Using the property \( a \log_b c = \log_b c^a \), we can rewrite the right side: \[ 5 \log_{4} 3 + \log_{4} 9 = \log_{4} 3^5 + \log_{4} 9 \] Combine the logarithms on the right side using the property \(\log_b a + \log_b c = \log_b (a \cdot c)\): \[ \log_{4} 3^5 + \log_{4} 9 = \log_{4} (3^5 \cdot 9) \]

Since the bases of the logarithms are the same, we equate the arguments: \[ (x+1)^2 = 3^5 \cdot 9 \] Calculate \(3^5 \cdot 9\): \[ 3^5 = 243 \quad \text{and} \quad 3^5 \cdot 9 = 243 \cdot 9 = 2187 \] Thus, the equation becomes: \[ (x+1)^2 = 2187 \]

Solve the equation \((x+1)^2 = 2187\) for \(x\): \[ x+1 = \pm \sqrt{2187} \] \[ x = -1 \pm \sqrt{2187} \] The exact solution is: \[ x = -1 + \sqrt{2187} \] The decimal approximation of the solution is: \[ x \approx 45.77 \]

Final Answer