Question
\( \rightarrow \log _ { 8 } ( b ^ { 2 } + 15 ) = 2 \)
Ask by Turner Daniels. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( b = 7 \) or \( b = -7 \).
Solution
We start with the equation:
log₈(b² + 15) = 2
Recall that the logarithmic equation logₐ(x) = y is equivalent to aʸ = x. Using this idea, we rewrite the equation as:
8² = b² + 15
Since 8² = 64, the equation becomes:
64 = b² + 15
Subtract 15 from both sides:
b² = 64 - 15
b² = 49
Taking the square root of both sides (and remembering to include both the positive and negative roots), we obtain:
b = 7 or b = -7
Both solutions are valid since b² + 15 is always positive for any real b, ensuring that the original logarithm is defined.
Thus, the solutions to the equation are:
b = 7 or b = -7.
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Simplify this solution Bonus Knowledge
To solve the equation \( \log _ { 8 } ( b ^ { 2 } + 15 ) = 2 \), you can start by rewriting it in exponential form. This gives you: \[ b^2 + 15 = 8^2. \] Calculating the right side, \( 8^2 = 64 \), the equation simplifies to: \[ b^2 + 15 = 64. \] Next, subtract 15 from both sides: \[ b^2 = 49. \] Taking the square root of both sides, you find: \[ b = 7 \quad \text{or} \quad b = -7. \] So the solutions to the equation are \( b = 7 \) and \( b = -7 \). Make sure to double-check those solutions by plugging them back into the original logarithmic equation to see if both satisfy it!
