What is the solution to $\log _4 8 x=3$?
Answer (1)
Rewrite the logarithmic equation in exponential form: 4 3 = 8 x .
Simplify the left side: 64 = 8 x .
Divide both sides by 8: x = 8 64 .
Simplify to find the value of x : x = 8 . The solution is 8 .
Explanation
Understanding the Problem We are given the equation lo g 4 8 x = 3 . Our goal is to find the value of x that satisfies this equation.
Converting to Exponential Form To solve for x , we first rewrite the logarithmic equation in exponential form. The equation lo g 4 8 x = 3 is equivalent to 4 3 = 8 x .
Simplifying the Equation Now, we simplify the left side of the equation. We know that 4 3 = 4 × 4 × 4 = 64 . So, our equation becomes 64 = 8 x .
Solving for x Next, we isolate x by dividing both sides of the equation by 8: 8 64 = 8 8 x 8 = x Thus, x = 8 .
Final Answer Therefore, the solution to the equation lo g 4 8 x = 3 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 of a solution in chemistry, and modeling population growth in biology. For example, if we know the intensity of an earthquake is 1000 times greater than the reference intensity, we can use logarithms to find its magnitude on the Richter scale: M = lo g 10 ( 1000 ) = 3 .
