Rewrite the following equation in logarithmic form.
[tex]$64=4^3$[/tex]
Rewrite the following equation in exponential form.
[tex]$\log _{49}(7)=\frac{1}{2}$[/tex]
Answer (2)
Rewrite 64 = 4 3 in logarithmic form using the definition of logarithm: lo g b y = x if b x = y .
Applying the definition, we get lo g 4 64 = 3 .
Rewrite lo g 49 ( 7 ) = 2 1 in exponential form using the definition of logarithm: b x = y if lo g b y = x .
Applying the definition, we get 4 9 2 1 = 7 .
The answers are lo g 4 64 = 3 and 4 9 2 1 = 7 .
Explanation
Understanding the Problem We are given two equations: one in exponential form and one in logarithmic form. Our task is to convert each equation to the other form.
Converting to Logarithmic Form The first equation is 64 = 4 3 . To rewrite this in logarithmic form, we use the definition of a logarithm: if b x = y , then lo g b y = x . In this case, b = 4 , x = 3 , and y = 64 . Therefore, the logarithmic form is lo g 4 64 = 3 .
Converting to Exponential Form The second equation is lo g 49 ( 7 ) = 2 1 . To rewrite this in exponential form, we again use the definition of a logarithm: if lo g b y = x , then b x = y . In this case, b = 49 , x = 2 1 , and y = 7 . Therefore, the exponential form is 4 9 2 1 = 7 .
Final Answer Therefore, the logarithmic form of 64 = 4 3 is lo g 4 64 = 3 , and the exponential form of lo g 49 ( 7 ) = 2 1 is 4 9 2 1 = 7 .
Examples
Logarithmic and exponential forms are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale or determining the pH levels of solutions in chemistry. For example, the Richter scale uses logarithms to measure the amplitude of seismic waves, where each whole number increase represents a tenfold increase in amplitude. Similarly, in finance, exponential functions are used to model compound interest, where the amount of money grows exponentially over time. Understanding these relationships helps in making informed decisions and predictions in these areas.
The logarithmic form of 64 = 4 3 is lo g 4 64 = 3 . The exponential form of lo g 49 ( 7 ) = 2 1 is 4 9 2 1 = 7 .
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