Using logarithmic properties, what is the solution to [tex]$\log _3(y+5)+\log _3 6=\log _3 66$[/tex]? Show all necessary steps.
Answer (2)
Use the logarithmic property lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) to combine the logarithms: lo g 3 ( 6 ( y + 5 )) = lo g 3 66 .
Equate the arguments of the logarithms since the bases are the same: 6 ( y + 5 ) = 66 .
Divide both sides by 6: y + 5 = 11 .
Solve for y : y = 11 − 5 = 6 . The solution is 6 .
Explanation
Understanding the Problem We are given the equation lo g 3 ( y + 5 ) + lo g 3 6 = lo g 3 66 . We need to find the value of y using logarithmic properties.
Combining Logarithms Using the logarithmic property lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) , we can combine the logarithms on the left side of the equation: lo g 3 ( y + 5 ) + lo g 3 6 = lo g 3 (( y + 5 ) × 6 ) So the equation becomes: lo g 3 ( 6 ( y + 5 )) = lo g 3 66
Equating Arguments Since the logarithms have the same base, we can equate the arguments: 6 ( y + 5 ) = 66
Dividing by 6 Divide both sides of the equation by 6: 6 6 ( y + 5 ) = 6 66 y + 5 = 11
Solving for y Subtract 5 from both sides to solve for y :
y + 5 − 5 = 11 − 5 y = 6
Checking the Solution Check the solution by substituting y = 6 back into the original equation to ensure that the arguments of the logarithms are positive. In this case, 0"> y + 5 = 6 + 5 = 11 > 0 , so the solution is valid. lo g 3 ( 6 + 5 ) + lo g 3 ( 6 ) = lo g 3 ( 11 ) + lo g 3 ( 6 ) = lo g 3 ( 11 × 6 ) = lo g 3 ( 66 ) The solution is correct.
Final Answer Therefore, the solution to the equation lo g 3 ( y + 5 ) + lo g 3 6 = lo g 3 66 is y = 6 .
Examples
Logarithmic properties are useful in many fields, including 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 compare the size of different earthquakes. Similarly, in finance, logarithmic scales are used to analyze stock market trends and investment growth over time. Understanding logarithmic properties helps in making informed decisions and predictions in these diverse areas.
To solve the equation lo g 3 ( y + 5 ) + lo g 3 6 = lo g 3 66 , we combine the logarithms to get lo g 3 ( 6 ( y + 5 )) = lo g 3 66 . Equating the arguments gives us 6 ( y + 5 ) = 66 , leading to y = 6 .
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