Express the equation in logarithmic form. Use ln rather than log.
(a) [tex]$e^x=6$[/tex] is equivalent to the logarithmic equation:
(b) [tex]$e^4=x$[/tex] is equivalent to the logarithmic equation:
Answer (2)
x = ln ( 6 ) ; 4 = ln ( x )
Explanation
Understanding the Problem We are given two exponential equations and asked to rewrite them in logarithmic form using the natural logarithm, denoted as 'ln'. The natural logarithm is the logarithm to the base e , where e is approximately 2.71828.
Converting the First Equation (a) The equation is e x = 6 . To express this in logarithmic form, we use the definition of the natural logarithm: if e y = z , then y = ln ( z ) . Applying this to our equation, we get x = ln ( 6 ) .
Converting the Second Equation (b) The equation is e 4 = x . Using the same definition, if e y = z , then y = ln ( z ) . In this case, we have 4 = ln ( x ) .
Final Answer Therefore, the logarithmic forms of the given equations are: (a) x = ln ( 6 ) (b) 4 = ln ( x )
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 example, the Richter scale uses logarithms to measure the amplitude of seismic waves, allowing scientists to quantify the energy released during an earthquake. Similarly, in finance, logarithmic scales are used to analyze investment growth and risk, providing a clearer picture of percentage changes over time.
The logarithmic forms of the equations are: (a) x = ln ( 6 ) and (b) 4 = ln ( x ) . These transformations employ the natural logarithm, which is the logarithm to the base e . This helps simplify exponential equations into a more manageable logarithmic format.
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