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)
Recognize the logarithmic form of the equation.
Apply the definition of logarithm to convert the equation to exponential form.
Identify the correct exponential form: x + 1 = 3 2 .
State the final answer: x + 1 = 3 2 .
Explanation
Understanding the problem We are given the logarithmic equation lo g 3 ( x + 1 ) = 2 and we need to find the equivalent exponential form.
Recalling the definition of logarithm Recall the definition of a logarithm: lo g b a = c is equivalent to b c = a . In our case, we have b = 3 , a = x + 1 , and c = 2 .
Converting to exponential form Applying the definition to the given equation, we get 3 2 = x + 1 , which can be rewritten as x + 1 = 3 2 .
Final Answer Therefore, the correct statement is x + 1 = 3 2 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring the acidity or alkalinity (pH) of a solution, and modeling population growth or decay. 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. Similarly, in chemistry, the pH of a solution is defined as the negative logarithm of the hydrogen ion concentration.
The logarithmic equation lo g 3 ( x + 1 ) = 2 translates to the exponential form x + 1 = 3 2 , which indicates that the correct statement is option A. Therefore, the statement that is true is x + 1 = 3 2 .
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