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 .
Simplify: x + 8 = 16 .
Solve for x : x = 8 .
Explanation
Understanding the Problem We are given the equation 4 l o g 4 ( x + 8 ) = 4 2 . Our goal is to solve for x .
Equating the Exponents Since the bases are the same, we can equate the exponents: lo g 4 ( x + 8 ) = 2 .
Rewriting in Exponential Form Now, we rewrite the equation in exponential form: x + 8 = 4 2 .
Simplifying the Equation Simplify the right side: x + 8 = 16 .
Isolating x Solve for x : x = 16 − 8 .
Calculating the Solution Calculate the value of x : x = 8 .
Verifying the Solution Check the solution: 0"> x + 8 > 0 , so 0"> 8 + 8 = 16 > 0 , which is true. Therefore, x = 8 is a valid solution.
Final Answer Thus, the solution to the equation 4 l o g 4 ( x + 8 ) = 4 2 is x = 8 .
Examples
Exponential and logarithmic equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to calculate compound interest. Imagine you invest 1000 inana cco u n tt ha tp a ys 5 t ye a rs i s A = 1000(1 + 0.05)^t$. Logarithms can then be used to determine how many years it will take for your investment to double.
