Question

Solve log8(x2−9)−log89=1 .(1 point) Responses x=±17−−√ x equals plus or minus Start Root 17 End Root x=±32–√ x equals plus or minus 3 Start Root 2 End Root x=9,−9 x equal 9 comma negative 9 x=9

Answer 1

To solve the equation \( \log_8(x^2 - 9) - \log_8(9) = 1 \), we can use the properties of logarithms.

1. Use the property of logarithms: \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \).

\[
\log_8\left(\frac{x^2 - 9}{9}\right) = 1
\]

2. Rewrite the equation using the definition of logarithms: If \( \log_b(a) = c \), then \( a = b^c \).

\[
\frac{x^2 - 9}{9} = 8^1
\]

This simplifies to:

\[
\frac{x^2 - 9}{9} = 8
\]

3. Multiply both sides by 9:

\[
x^2 - 9 = 72
\]

4. Add 9 to both sides:

\[
x^2 = 81
\]

5. Take the square root of both sides:

\[
x = \pm 9
\]

Thus, the solutions are \( x = 9 \) and \( x = -9 \).

The correct response is:

**x = 9, −9 (x equals 9, negative 9)**.