Which statement is true for $\log _3(x+1)=2$?
A. $x+1=3^2$
B. $x+1=2^3$
C. $2(x+1)=3$
D. $3(x+1)=2$
Answer (2)
Use the definition of logarithm to convert the equation from logarithmic form to exponential form.
The equation lo g 3 ( x + 1 ) = 2 is equivalent to x + 1 = 3 2 .
Compare the result with the given options.
The correct statement is x + 1 = 3 2 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 3 ( x + 1 ) = 2 . We need to find the equivalent exponential form of this equation from the given options.
Recalling the Definition of Logarithm Recall the definition of a logarithm: lo g b a = c is equivalent to b c = a . In other words, the base b raised to the power of c equals a .
Converting to Exponential Form Applying this definition to our equation lo g 3 ( x + 1 ) = 2 , we identify the base as b = 3 , the exponent as c = 2 , and the argument of the logarithm as a = x + 1 . Therefore, the equivalent exponential form is 3 2 = x + 1 , which can be rewritten as x + 1 = 3 2 .
Identifying the Correct Statement Comparing this result with the given options, we see that the first option, x + 1 = 3 2 , matches our derived equation.
Examples
Logarithms are used to solve exponential equations, which appear in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth or radioactive decay. For example, if we know the population doubles every 10 years and we want to know how long it takes to triple, we can use logarithms to solve the exponential equation that models this growth.
The logarithmic equation lo g 3 ( x + 1 ) = 2 can be rewritten as x + 1 = 3 2 . Therefore, the correct answer is option A: x + 1 = 3 2 . This aligns with the definition of logarithms converted to exponential form.
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