$\log _3(x+1)-\log _3 x=4$
Select the correct choice below and, if necessary, fill in the answer
A. The solution is $x=$ $\square$ .
(Simplify your answer. Type an integer or a fraction. Use a c
B. The solution is not a real number.
Answer (1)
Use the logarithm property to combine the logarithms: lo g 3 x x + 1 = 4 .
Convert the logarithmic equation to exponential form: x x + 1 = 3 4 = 81 .
Solve for x : x + 1 = 81 x ⇒ 1 = 80 x ⇒ x = 80 1 .
The solution is x = 80 1 .
Explanation
Understanding the Problem We are given the equation lo g 3 ( x + 1 ) − lo g 3 x = 4 . Our goal is to solve for x and determine if the solution is a real number. The domain of the logarithm function requires its argument to be positive, so we must have 0"> x + 1 > 0 and 0"> x > 0 , which means 0"> x > 0 .
Applying Logarithm Properties Using the logarithm property lo g a b − lo g a c = lo g a c b , we can rewrite the equation as lo g 3 x x + 1 = 4 .
Converting to Exponential Form Now, we convert the logarithmic equation to an exponential equation: x x + 1 = 3 4 .
Simplifying the Equation Simplify the equation: x x + 1 = 81 .
Solving for x Solve for x : x + 1 = 81 x .
Isolating x Isolate x : 1 = 81 x − x , which simplifies to 1 = 80 x .
Finding the Solution Solve for x : x = 80 1 .
Checking the Domain Check if the solution is in the domain of the logarithmic functions. Since 0"> x = 80 1 > 0 , the solution is valid.
Final Answer Therefore, the solution is x = 80 1 .
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 of a solution in chemistry. For example, if we know the ratio of the amplitudes of seismic waves from two earthquakes and want to find the difference in their magnitudes, we can use logarithmic equations to solve for the unknown magnitude. This allows us to compare the relative sizes of earthquakes and understand the energy they release.
