What is the solution to [tex]$\log 25 x=3$[/tex]?
A. [tex]$x=\frac{3}{25}$[/tex]
B. [tex]$x=\frac{8}{25}$[/tex]
C. [tex]$x=35$[/tex]
D. [tex]$x=40$[/tex]
Answer (2)
Rewrite the logarithmic equation in exponential form: 25 x = 1 0 3 .
Simplify the right side: 25 x = 1000 .
Divide both sides by 25: x = 25 1000 .
Simplify the fraction: x = 40 . The solution is 40 .
Explanation
Understanding the Problem We are given the equation lo g 25 x = 3 . The base of the logarithm is not specified, so we assume it is base 10. We need to solve for x .
Converting to Exponential Form To solve the equation, we first rewrite the logarithmic equation in exponential form. Since the base is 10, we have 25 x = 1 0 3 .
Simplifying the Equation Next, we simplify the right side of the equation: 1 0 3 = 1000 , so we have 25 x = 1000 .
Isolating x Now, we divide both sides of the equation by 25 to isolate x : x = 25 1000 .
Finding the Value of x Finally, we simplify the fraction to find the value of x : x = 40 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH). Understanding how to solve logarithmic equations is crucial in these fields. For example, if we know the intensity of an earthquake is 1000 times greater than the reference intensity, we can use logarithms to determine its magnitude on the Richter scale: M = lo g 10 ( 1000 ) = 3 .
The solution to the equation lo g 10 ( 25 x ) = 3 is obtained by converting it to exponential form, leading to x = 40 . Therefore, the answer is option D: x = 40 .
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