Convert to an exponential equation. [tex]$\log _n M=-y$[/tex]
Answer (2)
The problem requires converting a logarithmic equation to its equivalent exponential form.
Recall the definition of logarithm: lo g b a = c is equivalent to b c = a .
Identify the base, argument, and logarithm value in the given equation.
Rewrite the equation in exponential form: n − y = M .
Explanation
Understanding the Problem We are given the logarithmic equation lo g n M = − y and we want to convert it to its equivalent exponential form.
Recalling the Definition of Logarithm Recall that a logarithm is the inverse operation to exponentiation. The logarithmic equation lo g b a = c is equivalent to the exponential equation b c = a . In other words, the base b raised to the power of c gives us a .
Identifying the Components In our case, we have lo g n M = − y . Comparing this to the general form lo g b a = c , we can identify the base as b = n , the argument of the logarithm as a = M , and the logarithm value as c = − y .
Converting to Exponential Form Now, we can rewrite the logarithmic equation in exponential form using the relationship b c = a . Substituting the values we identified, we get n − y = M .
Final Answer Therefore, the exponential form of the given logarithmic equation is n − y = M .
Examples
Logarithmic scales are used to represent large ranges of values in a more manageable way. For example, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An increase of one unit on the Richter scale corresponds to a tenfold increase in the amplitude of the earthquake waves. Similarly, the decibel scale, used to measure sound intensity, is also logarithmic. Converting between logarithmic and exponential forms helps in understanding and comparing these measurements.
To convert the logarithmic equation lo g n M = − y into exponential form, you rewrite it as n − y = M . This shows that the base raised to the negative exponent equals the argument. Thus, the final answer is n − y = M .
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