Which graph shows the solution to the equation $\log _2(3 x-1)=2 $?

Rewrite the logarithmic equation in exponential form: 3 x − 1 = 2 2 .
Simplify the equation: 3 x − 1 = 4 .
Solve for x : 3 x = 5 , which gives x = 3 5 ​ .
The solution is x = 3 5 ​ , represented graphically as a point on the number line: 3 5 ​ ​ .

Explanation

Understanding the Problem We are given the equation lo g 2 ​ ( 3 x − 1 ) = 2 and asked to find its solution and represent it graphically.

Converting to Exponential Form To solve the logarithmic equation, we first rewrite it in exponential form. The equation lo g 2 ​ ( 3 x − 1 ) = 2 is equivalent to 3 x − 1 = 2 2 .

Simplifying the Equation Now, we simplify the equation: 3 x − 1 = 4 .

Solving for x Next, we solve for x . Adding 1 to both sides gives 3 x = 5 . Dividing both sides by 3, we find x = 3 5 ​ .

Graphical Representation The solution to the equation is x = 3 5 ​ . This can be represented graphically as a single point on the number line at x = 3 5 ​ .


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 .

Answered by GinnyAnswer | 2025-07-05