Write the equation in its equivalent logarithmic form. [tex]4^2=16[/tex]
Answer (2)
The equation is given in exponential form: 4 2 = 16 .
Recall the definition of logarithm: If b x = y , then lo g b y = x .
Apply this definition to the given equation 4 2 = 16 . Here, b = 4 , x = 2 , and y = 16 .
The equivalent logarithmic form is lo g 4 16 = 2 .
Explanation
Understanding the Problem The problem gives us an equation in exponential form, 4 2 = 16 , and asks us to rewrite it in its equivalent logarithmic form. To do this, we need to understand the relationship between exponential and logarithmic forms.
Recalling the Definition of Logarithm Recall that the exponential form b x = y is equivalent to the logarithmic form lo g b y = x . In other words, the logarithm (base b ) of y is the exponent x to which we must raise b to obtain y .
Applying the Definition to the Given Equation In our equation, 4 2 = 16 , we can identify the base b = 4 , the exponent x = 2 , and the result y = 16 . Applying the definition of logarithm, we can rewrite the equation as lo g 4 16 = 2 . This means that the logarithm base 4 of 16 is 2, because 4 2 = 16 .
Final Answer Therefore, the equivalent logarithmic form of the equation 4 2 = 16 is lo g 4 16 = 2 .
Examples
Logarithmic equations are used in many real-world applications, such as calculating the magnitude of earthquakes using the Richter scale, determining the loudness of sound using decibels, and modeling population growth. For example, if we know that the population of a town doubles every 10 years, we can use logarithms to determine how many years it will take for the population to increase by a certain factor.
The equation 4 2 = 16 can be rewritten in logarithmic form as lo g 4 16 = 2 . This means that 4 raised to the power of 2 equals 16. Therefore, the base 4 logarithm of 16 is 2.
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