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 (2)
Apply the property a l o g a ( x ) = x to simplify the equation to x + 8 = 4 2 .
Calculate 4 2 to get x + 8 = 16 .
Subtract 8 from both sides to solve for x .
The solution is x = 8 .
Explanation
Understanding the Problem 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.
Applying Logarithmic Properties We can use the property of logarithms that states a l o g a ( b ) = b . Applying this property to the left side of the equation, we get x + 8 = 4 2 .
Calculating the Power Now we need to calculate 4 2 . 4 2 = 4 × 4 = 16 . So, our equation becomes x + 8 = 16 .
Solving for x To solve for x , we subtract 8 from both sides of the equation: x = 16 − 8 . Therefore, x = 8 .
Final Answer The solution to the equation 4 l o g 4 ( x + 8 ) = 4 2 is x = 8 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH levels in chemistry, and modeling population growth in biology. Understanding how to solve these equations allows us to analyze and interpret data in these real-world scenarios.
The solution to the equation 4 l o g 4 ( x + 8 ) = 4 2 is x = 8 .
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