A) i) If today is Tuesday, what day will it be after 200 days?
ii) If [tex]$\log (5 x-4)=\log (x+1)+\log 4$[/tex], find the value of [tex]$x$[/tex]
B) Express [tex]$\frac{1}{\sqrt{5}}+\sqrt{20}$[/tex] in the form [tex]$a \sqrt{b}$[/tex]
C) If [tex]$2^x+2^{(x-1)}=48$[/tex], find the value of [tex]$x$[/tex]
Answer (2)
Determine the day of the week after 200 days: 200 ≡ 4 ( mod 7 ) , so it will be Saturday.
Solve the logarithmic equation: lo g ( 5 x − 4 ) = lo g ( x + 1 ) + lo g 4 ⟹ 5 x − 4 = 4 ( x + 1 ) ⟹ x = 8 .
Express 5 1 + 20 in the form a b : 5 1 + 20 = 5 5 + 2 5 = 5 11 5 .
Solve the exponential equation: 2 x + 2 x − 1 = 48 ⟹ 2 x + 2 2 x = 48 ⟹ 2 x = 32 ⟹ x = 5 .
x = 5
Explanation
Finding the Remainder We need to determine the day of the week 200 days from Tuesday. There are 7 days in a week, so we can find the remainder when 200 is divided by 7.
Calculating the Day The remainder when 200 is divided by 7 is 4, since 200 = 7 × 28 + 4 . This means that after 200 days, it will be 4 days after Tuesday.
Determining the Day of the Week Counting 4 days from Tuesday, we have Wednesday, Thursday, Friday, and Saturday. Therefore, it will be Saturday after 200 days.
Simplifying the Logarithmic Equation We are given the equation lo g ( 5 x − 4 ) = lo g ( x + 1 ) + lo g 4 . We need to find the value of x . Using the properties of logarithms, we can combine the terms on the right side.
Combining Logarithms Using the property lo g a + lo g b = lo g ( ab ) , we have lo g ( x + 1 ) + lo g 4 = lo g [ 4 ( x + 1 )] . So the equation becomes lo g ( 5 x − 4 ) = lo g [ 4 ( x + 1 )] .
Equating Arguments Since the logarithms are equal, their arguments must be equal. Thus, 5 x − 4 = 4 ( x + 1 ) .
Solving for x Expanding the right side, we get 5 x − 4 = 4 x + 4 . Subtracting 4 x from both sides gives x − 4 = 4 . Adding 4 to both sides gives x = 8 .
Simplifying the Expression We need to express 5 1 + 20 in the form a b . First, we simplify 20 .
Simplifying the Square Root Since 20 = 4 × 5 , we have 20 = 4 × 5 = 4 × 5 = 2 5 . So the expression becomes 5 1 + 2 5 .
Rationalizing the Denominator To rationalize the denominator of 5 1 , we multiply the numerator and denominator by 5 , which gives 5 1 = 5 5 .
Combining Terms Now the expression is 5 5 + 2 5 . We can write 2 5 as 5 10 5 . So the expression becomes 5 5 + 5 10 5 = 5 11 5 .
Final Form Thus, the expression 5 1 + 20 can be written as 5 11 5 , which is in the form a b , where a = 5 11 and b = 5 .
Rewriting the Exponential Equation We are given the equation 2 x + 2 ( x − 1 ) = 48 . We need to find the value of x . We can rewrite 2 ( x − 1 ) as 2 2 x .
Factoring So the equation becomes 2 x + 2 2 x = 48 . Factoring out 2 x , we get 2 x ( 1 + 2 1 ) = 48 , which simplifies to 2 x ( 2 3 ) = 48 .
Isolating 2^x Multiplying both sides by 3 2 , we get 2 x = 48 × 3 2 = 16 × 2 = 32 .
Solving for x Since 32 = 2 5 , we have 2 x = 2 5 . Therefore, x = 5 .
Final Answers A) i) The day after 200 days from Tuesday is Saturday. ii) The value of x in the logarithmic equation is 8. B) The expression 5 1 + 20 in the form a b is 5 11 5 .
C) The value of x in the exponential equation is 5.
Examples
Understanding days of the week is crucial in scheduling and planning events. For instance, if a project starts on a Tuesday and requires 200 days to complete, knowing that the completion day will be a Saturday helps in coordinating resources and setting realistic deadlines. Logarithmic equations are used in various fields like finance (calculating interest rates) and science (measuring acidity). Simplifying radical expressions is useful in engineering and physics for calculations involving distances and areas. Exponential equations are used in modeling population growth and radioactive decay, helping scientists make predictions and informed decisions.
After 200 days from Tuesday, it will be Saturday. The value of x in the logarithmic equation is 8. The expression over sqrt(5) + sqrt(20) can be rewritten as 1sqrt(5)/5, and the value of x in the exponential equation is 5.
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