Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. log 3x=3
Answer (2)
Rewrite the logarithmic equation in exponential form: 1 0 3 = 3 x .
Solve for x : x = 3 1000 .
Check if the solution is in the domain of the original logarithmic expression: 0"> x > 0 .
The exact answer is 3 1000 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g ( 3 x ) = 3 . Our goal is to solve for x , making sure that our solution is within the domain of the original logarithmic expression. Remember, the domain of a logarithmic function lo g ( u ) is 0"> u > 0 . Therefore, we must have 0"> 3 x > 0 , which means 0"> x > 0 .
Converting to Exponential Form To solve the equation, we need to rewrite it in exponential form. The base of the logarithm is not explicitly written, so we assume it to be 10. Using the definition of logarithms, if lo g b ( a ) = c , then b c = a . In our case, lo g ( 3 x ) = 3 , so 1 0 3 = 3 x .
Solving for x Now we solve the equation 1 0 3 = 3 x for x . We have 1000 = 3 x . Dividing both sides by 3, we get x = 3 1000 .
Checking the Domain We need to check if our solution is in the domain of the original logarithmic expression. Since 0"> x = 3 1000 > 0 , the solution is valid.
Final Answer Therefore, the exact answer for x is 3 1000 .
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 instance, if we know the intensity of an earthquake is 1000 times greater than the reference intensity, we can use a logarithmic scale to express its magnitude as log(1000) = 3. This makes it easier to compare the relative sizes of different earthquakes.
To solve lo g ( 3 x ) = 3 , we convert it to exponential form to get 1 0 3 = 3 x and then solve for x to find x = 3 1000 . This solution is valid since it satisfies the domain condition of the logarithm. Thus, the exact answer is 3 1000 .
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