Solve the following logarithmic equation for [tex]$x$\\\log _z 81=4$[/tex]
[tex]$x =81$[/tex]
[tex]$x=9$[/tex]
[tex]$x=4$[/tex]
[tex]$x=3$[/tex]
Use the given formula to determine the answer [tex]$R=\\log \\left(\frac{A}{A_0}\\right)$[/tex]
An earthquake monitoring station measured the amplitude of the waves during a recent tremor. The waves were 10,000 times as large as [tex]$A_0$[/tex], the smallest detectable wave. How high did this earthquake measure on the Richter scale?
4
10,000
10
8
5
Answer (2)
Test each possible value of x in the equation x ! lo g z 81 = 4 and find that x = 3 , 4 , 9 , 81 are all possible solutions.
Substitute A = 10000 A 0 into the Richter scale formula R = lo g ( A 0 A ) .
Simplify the expression to find R = lo g ( 10000 ) .
Calculate the logarithm to find the Richter scale measurement: 4 .
Explanation
Analyzing the Logarithmic Equation We are given the logarithmic equation x ! lo g z 81 = 4 and the possible values for x are 3 , 4 , 9 , 81 . We need to find which of these values satisfy the equation for some z .
Testing Possible Values of x Let's test each possible value of x .
If x = 3 , then 3 ! lo g z 81 = 6 lo g z 81 = 4 . This implies lo g z 81 = 6 4 = 3 2 , so z 3 2 = 81 , which means z = 8 1 2 3 = ( 9 2 ) 2 3 = 9 3 = 729 . Thus, x = 3 is a possible solution.
If x = 4 , then 4 ! lo g z 81 = 24 lo g z 81 = 4 . This implies lo g z 81 = 24 4 = 6 1 , so z 6 1 = 81 , which means z = 8 1 6 = ( 3 4 ) 6 = 3 24 . Thus, x = 4 is a possible solution.
If x = 9 , then 9 ! lo g z 81 = 4 . This implies lo g z 81 = 9 ! 4 , so z 9 ! 4 = 81 , which means z = 8 1 4 9 ! . Thus, x = 9 is a possible solution.
If x = 81 , then 81 ! lo g z 81 = 4 . This implies lo g z 81 = 81 ! 4 , so z 81 ! 4 = 81 , which means z = 8 1 4 81 ! . Thus, x = 81 is a possible solution.
Calculating Richter Scale Measurement Now, let's analyze the Richter scale problem. We are given the formula R = lo g ( A 0 A ) , where A = 10000 A 0 . We need to find the value of R .
Substituting A = 10000 A 0 into the formula, we get: R = lo g ( A 0 10000 A 0 ) = lo g ( 10000 ) = lo g ( 1 0 4 ) = 4 .
Final Answer Therefore, the Richter scale measurement is 4.
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. The Richter scale, for example, uses logarithms to measure the amplitude of seismic waves, allowing scientists to quantify the energy released during an earthquake. Understanding logarithmic scales helps in interpreting and comparing vastly different magnitudes of events in a meaningful way.
The solution to the logarithmic equation lo g z 81 = 4 is x = 3 . For the Richter scale calculation, the earthquake measured 4, as derived from the formula R = lo g ( 10000 ) . Thus, the answer for both questions confirms that x = 3 and the Richter scale reading is 4.
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