Solve the logarithmic equation.
$\log _4 x=3$
$x=$
(Simplify your answer. Type an exact answer, using $e$ as needed.)
Answer (1)
64
Explanation
Understanding the Problem We are given the logarithmic equation lo g 4 x = 3 . Our goal is to find the value of x that satisfies this equation.
Converting to Exponential Form To solve for x , we need to rewrite the logarithmic equation in its equivalent exponential form. Recall that the logarithmic equation lo g b a = c is equivalent to the exponential equation b c = a . Applying this to our equation, we have 4 3 = x .
Calculating the Value of x Now, we simply calculate 4 3 . This means 4 × 4 × 4 = 16 × 4 = 64 . Therefore, x = 64 .
Final Answer Thus, the solution to the logarithmic equation lo g 4 x = 3 is x = 64 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH levels of chemical solutions. 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 . This shows how logarithms help us quantify and understand phenomena that vary over a wide range of values.
