If [tex]$\log _m 9=2$[/tex], then [tex]$m=3$[/tex].
True
False
Answer (2)
Rewrite the logarithmic equation in exponential form: m 2 = 9 .
Solve for m : m = ± 3 .
Since the base of a logarithm must be positive, m = − 3 is not a valid solution.
Therefore, the statement is true: T r u e
Explanation
Understanding the Problem We are given the equation lo g m 9 = 2 and asked to determine if m = 3 . The base of a logarithm must be positive and not equal to 1.
Converting to Exponential Form To solve this, we can rewrite the logarithmic equation in exponential form. The equation lo g m 9 = 2 is equivalent to m 2 = 9 .
Solving for m Now, we solve for m by taking the square root of both sides of the equation m 2 = 9 . This gives us m = ± 3 .
Checking the Validity of the Solution However, the base of a logarithm must be positive, so m = − 3 is not a valid solution. Also, the base cannot be 1. Therefore, m = 3 is a valid solution.
Conclusion Thus, the statement 'If lo g m 9 = 2 , then m = 3 ' is true.
Examples
Logarithms are used in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring the loudness of sound in decibels, and determining the pH of a solution in chemistry. Understanding how to solve logarithmic equations is essential for these applications. For example, if we know the magnitude of an earthquake is 7 on the Richter scale, we can use logarithms to determine the amplitude of the seismic waves. Similarly, in finance, logarithms are used to calculate compound interest and analyze investment growth.
The logarithmic equation lo g m 9 = 2 implies m 2 = 9 , which gives solutions m = ± 3 . Since the base of a logarithm must be positive, the valid solution is m = 3 . Therefore, the statement is true.
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