Question

Solve the logarithmic equation. Expre log₂(x²-9)-log₂(x+3)=1

Answer 1

The solution to the logarithmic equation log₂(x²-9) - log₂(x+3) = 1 is x = 5.

Step-by-step explanation:

To solve the given logarithmic equation, we can use logarithmic properties to simplify and isolate the variable. Start by applying the quotient rule for logarithms, which states that log(a) - log(b) = log(a/b). In this case, we can rewrite the equation as log₂((x²-9)/(x+3)) = 1.

Next, use the definition of logarithms to convert the equation into exponential form. The base 2 raised to the power of 1 is 2, so we have 2 = (x²-9)/(x+3). Now, cross-multiply and simplify the resulting quadratic equation.

2(x+3) = x²-9.

Expand and rearrange the terms to obtain a quadratic equation in standard form:

x² - 2x - 15 = 0.

Factor the quadratic equation:

(x-5)(x+3) = 0.

Set each factor equal to zero and solve for x:

`\[x-5 = 0 \implies x = 5\]`

`\[x+3 = 0 \implies x = -3.\]`

However, we need to check for extraneous solutions. When x = -3, the denominator of the original logarithmic expression becomes zero, making the logarithm undefined. Thus, the only valid solution is x = 5.