Question 21 (5 points)
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By converting to an exponential expression, solve $\log _2(x+5)=4$
$x=-6$
$x=16$
$x=11$
$x=5$
Question 22 (5 points)
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Is $x=0$ a valid potential solution to the equation $\log (x+2)+\log (x+5)=1$ ?
Answer (1)
Solve lo g 2 ( x + 5 ) = 4 by converting to exponential form: 2 4 = x + 5 .
Simplify to 16 = x + 5 , then solve for x : x = 16 − 5 = 11 .
Substitute x = 0 into lo g ( x + 2 ) + lo g ( x + 5 ) = 1 to get lo g ( 2 ) + lo g ( 5 ) = 1 .
Simplify using logarithm rules: lo g ( 10 ) = 1 , confirming x = 0 is a valid solution.
Explanation
Convert to Exponential Form We are given the equation lo g 2 ( x + 5 ) = 4 . To solve for x , we need to convert this logarithmic equation into its equivalent exponential form.
Write Exponential Equation The exponential form of the equation lo g 2 ( x + 5 ) = 4 is 2 4 = x + 5 .
Calculate 2 to the power of 4 We know that 2 4 = 16 , so we have 16 = x + 5 .
Isolate x To solve for x , we subtract 5 from both sides of the equation: x = 16 − 5 .
Solve for x Therefore, x = 11 .
Analyze the Second Question Now, let's consider the second question. We are given the equation lo g ( x + 2 ) + lo g ( x + 5 ) = 1 , and we want to determine if x = 0 is a valid potential solution.
Substitute x=0 Substitute x = 0 into the equation: lo g ( 0 + 2 ) + lo g ( 0 + 5 ) = lo g ( 2 ) + lo g ( 5 ) .
Apply Logarithm Product Rule Using the logarithm product rule, which states that lo g ( a ) + lo g ( b ) = lo g ( ab ) , we can simplify the expression: lo g ( 2 ) + lo g ( 5 ) = lo g ( 2 × 5 ) = lo g ( 10 ) .
Evaluate Logarithm Since the base of the logarithm is 10, lo g 10 ( 10 ) = 1 . This is equal to the right-hand side of the equation, which is 1.
Conclusion Therefore, x = 0 is a valid potential solution to the equation lo g ( x + 2 ) + lo g ( x + 5 ) = 1 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution in chemistry, and modeling population growth in biology. For example, if we know the intensity of an earthquake is 1000 times greater than the reference intensity, we can use logarithms to find its magnitude on the Richter scale: M = lo g 10 ( 1000 ) = 3 . This shows how logarithms help simplify and quantify large-scale phenomena.
