Questions: Solve the following equation analytically. (Enter your answers as a com log4(x-3)+log4(x+3)=2 x=

Solution Steps

To solve the given logarithmic equation analytically, we can use the properties of logarithms to combine the terms and then solve for \( x \).

1. Use the product rule of logarithms: \(\log_b(a) + \log_b(c) = \log_b(ac)\).
2. Set the combined logarithm equal to the given value.
3. Convert the logarithmic equation to its exponential form.
4. Solve the resulting quadratic equation for \( x \).

We start with the equation: \[ \log_{4}(x - 3) + \log_{4}(x + 3) = 2 \] Using the product rule of logarithms, we can combine the left side: \[ \log_{4}((x - 3)(x + 3)) = 2 \]

Next, we convert the logarithmic equation to its exponential form: \[ (x - 3)(x + 3) = 4^2 \] This simplifies to: \[ (x - 3)(x + 3) = 16 \]

Expanding the left side gives us: \[ x^2 - 9 = 16 \] Rearranging this leads to: \[ x^2 - 25 = 0 \]

Factoring the quadratic equation, we have: \[ (x - 5)(x + 5) = 0 \] Thus, the solutions are: \[ x = 5 \quad \text{or} \quad x = -5 \]

Since we are dealing with logarithms, we need to ensure that the arguments of the logarithms are positive:

• For \(x = 5\): \(x - 3 = 2 > 0\) and \(x + 3 = 8 > 0\) (valid)
• For \(x = -5\): \(x - 3 = -8 < 0\) and \(x + 3 = -2 < 0\) (invalid)

Thus, the only valid solution is \(x = 5\).

Final Answer