What is the solution to $4^{\log _4(x+8)}=4^2$?
A. $x=-8$
B. $x=-4$
C. $x=4$
D. $x=8$
Answer (1)
Equate the exponents: lo g 4 ( x + 8 ) = 2 .
Rewrite in exponential form: x + 8 = 4 2 .
Solve for x : x = 4 2 − 8 .
Calculate x : x = 8 . The solution is 8 .
Explanation
Problem Analysis We are given the equation 4 l o g 4 ( x + 8 ) = 4 2 . Our goal is to find the value of x that satisfies this equation.
Equating Exponents Since the bases are the same, we can equate the exponents: lo g 4 ( x + 8 ) = 2 .
Exponential Form Now, we rewrite the equation in exponential form: x + 8 = 4 2 .
Isolating x Next, we solve for x : x = 4 2 − 8 .
Calculating x Finally, we calculate the value of x : x = 16 − 8 = 8 . Therefore, the solution to the equation is x = 8 .
Examples
Imagine you are adjusting the settings on a sound equalizer. The equation we solved is similar to how the equalizer adjusts sound frequencies. By understanding exponential and logarithmic relationships, you can fine-tune audio settings to achieve the perfect sound balance. This skill is useful not only in audio engineering but also in fields like data analysis, where logarithmic scales help visualize and interpret large datasets.
