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