Answer :
Let's solve the equation [tex]\(4^{\log_4(x+8)} = 4^2\)[/tex] step-by-step.
1. Recognize the Exponentiation Property:
Given the equation [tex]\(4^{\log_4(x+8)} = 4^2\)[/tex], notice that both sides of the equation have the same base, which is 4. When bases are the same, we can equate the exponents.
2. Equate the Exponents:
[tex]\[ \log_4(x + 8) = 2 \][/tex]
3. Expression in Logarithmic Form:
The equation [tex]\(\log_4(x + 8) = 2\)[/tex] implies that [tex]\(4\)[/tex] raised to the power of [tex]\(2\)[/tex] equals [tex]\(x + 8\)[/tex].
4. Convert to Exponential Form:
[tex]\[ x + 8 = 4^2 \][/tex]
5. Evaluate the Exponential Expression:
Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
6. Solve for [tex]\( x \)[/tex]:
Now, set up the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ x + 8 = 16 \][/tex]
Subtract 8 from both sides:
[tex]\[ x = 16 - 8 \][/tex]
[tex]\[ x = 8 \][/tex]
Therefore, the solution to the equation [tex]\(4^{\log_4(x+8)} = 4^2\)[/tex] is [tex]\( x = 8 \)[/tex].
Among the provided choices, the correct answer is:
[tex]\[ \boxed{8} \][/tex]
1. Recognize the Exponentiation Property:
Given the equation [tex]\(4^{\log_4(x+8)} = 4^2\)[/tex], notice that both sides of the equation have the same base, which is 4. When bases are the same, we can equate the exponents.
2. Equate the Exponents:
[tex]\[ \log_4(x + 8) = 2 \][/tex]
3. Expression in Logarithmic Form:
The equation [tex]\(\log_4(x + 8) = 2\)[/tex] implies that [tex]\(4\)[/tex] raised to the power of [tex]\(2\)[/tex] equals [tex]\(x + 8\)[/tex].
4. Convert to Exponential Form:
[tex]\[ x + 8 = 4^2 \][/tex]
5. Evaluate the Exponential Expression:
Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
6. Solve for [tex]\( x \)[/tex]:
Now, set up the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ x + 8 = 16 \][/tex]
Subtract 8 from both sides:
[tex]\[ x = 16 - 8 \][/tex]
[tex]\[ x = 8 \][/tex]
Therefore, the solution to the equation [tex]\(4^{\log_4(x+8)} = 4^2\)[/tex] is [tex]\( x = 8 \)[/tex].
Among the provided choices, the correct answer is:
[tex]\[ \boxed{8} \][/tex]
