Solve for $x$.
$3500=700 \cdot 2^{8 x}$
Check all that apply.
A. $x=\frac{\ln (5)}{8 \cdot \ln (2)}$
B. $x=\frac{\log (5)}{8}$
C. $x=\frac{\log (5)}{8 \cdot \log (2)}$
D. $x=\frac{\ln (8)}{5}
Answer (2)
Divide both sides of the equation by 700 to get 5 = 2 8 x .
Take the natural logarithm of both sides: ln ( 5 ) = ln ( 2 8 x ) .
Apply the power rule of logarithms and solve for x : x = 8 ⋅ l n ( 2 ) l n ( 5 ) .
Alternatively, take the common logarithm of both sides and solve for x : x = 8 ⋅ l o g ( 2 ) l o g ( 5 ) .
The correct options are A and C: x = 8 ⋅ ln ( 2 ) ln ( 5 ) , x = 8 ⋅ lo g ( 2 ) lo g ( 5 ) .
Explanation
Problem Setup We are given the equation 3500 = 700"." 2 8 x and asked to solve for x . We need to identify which of the given options are correct.
Simplify the Equation First, divide both sides of the equation by 700: 700 3500 = 2 8 x 5 = 2 8 x
Apply Natural Logarithm Now, take the logarithm of both sides. We can use either the natural logarithm ( ln ) or the common logarithm ( lo g ). Using the natural logarithm: ln ( 5 ) = ln ( 2 8 x )
Use Power Rule of Logarithms Apply the power rule of logarithms: ln ( a b ) = b ln ( a ) ln ( 5 ) = 8 x ⋅ ln ( 2 )
Solve for x (Natural Logarithm) Solve for x :
x = 8 ⋅ ln ( 2 ) ln ( 5 )
Apply Common Logarithm Using the common logarithm: lo g ( 5 ) = lo g ( 2 8 x )
Use Power Rule of Logarithms Apply the power rule of logarithms: lo g ( a b ) = b lo g ( a ) lo g ( 5 ) = 8 x ⋅ lo g ( 2 )
Solve for x (Common Logarithm) Solve for x :
x = 8 ⋅ lo g ( 2 ) lo g ( 5 )
Compare with Given Options Comparing the derived solutions with the given options: Option A: x = 8 ⋅ l n ( 2 ) l n ( 5 ) - This matches our solution using the natural logarithm. Option B: x = 8 l o g ( 5 ) - This does not match our solution. Option C: x = 8 ⋅ l o g ( 2 ) l o g ( 5 ) - This matches our solution using the common logarithm. Option D: x = 5 l n ( 8 ) - This does not match our solution.
Final Answer Therefore, the correct options are A and C.
Examples
Logarithmic functions are incredibly useful in various real-world scenarios. For instance, they help in calculating the time it takes for an investment to double at a certain interest rate. Suppose you invest 1000 a t anann u a l in t eres t r a t eo f 5 t$ it takes for the investment to double to $2000 is given by 2000 = 1000 e 0.05 t . Solving for t involves logarithms: 2 = e 0.05 t , so ln ( 2 ) = 0.05 t , and t = 0.05 l n ( 2 ) . Logarithms are also used in measuring the intensity of earthquakes (Richter scale) and sound levels (decibels).
To solve the equation 3500 = 700 ⋅ 2 8 x , we derive the appropriate formulas using both natural and common logarithms. The correct answers are options A and C: x = 8 ⋅ l n ( 2 ) l n ( 5 ) and x = 8 ⋅ l o g ( 2 ) l o g ( 5 ) .
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