Please check answers:
Use the equation of the exponential function whose graph passes through the points (0,-2) and (2,-50) to find the value of y when x= -2.
My answer: -2/25
Solve 64^x<32^x+2
Answer: x<10
Write the equation log(243)81=4/5
Answer: 243^4/5 = 81
Evaluate 9^log954
Answer: 6
Solve log(1/8)x = -1
Answer: 8
Solve log2(7x-3)>/= log2(x+12)
Answer: x>/=5/2
Thanks.
<<Solve 64^x < 32^(x+2) >>
64^x < 32^(x+2) = 32
(32*2)^x < 32^x * 32^2
32^x*2^x < 32<x * 32^2
2^x < 32^2 = 1024 = 2^10
x < 10
Good job!
<<Solve log(1/8)x = -1 >>
(1/8)^-1 = x
8 = x
Please check answers:
1. Use the equation of the exponential function whose graph passes through the points (0,-2) and (2,-50) to find the value of y when x= -2.
My answer: -2/25
Correct!
2. Solve 64^x<32^x+2
Answer: x<10
Correct!
3. Write the equation log(243)81=4/5
Answer: 243^4/5 = 81
Correct!
4. Evaluate 9^log954
Answer: 6
Correct!
5. Solve log(1/8)x = -1
Answer: 8
Incorrect. The correct answer is x=3.
6. Solve log2(7x-3)>/= log2(x+12)
Answer: x>/=5/2
Incorrect. The correct answer is x >= 3.
Overall, most of your answers are correct, but some need to be revised.
Let's check the answers one by one:
1. Use the equation of the exponential function whose graph passes through the points (0, -2) and (2, -50) to find the value of y when x = -2.
To find the equation of an exponential function, we use the general form: y = ab^x.
First, substitute the given points into the equation to get two equations:
-2 = ab^0 and -50 = ab^2.
Since any number (except zero) raised to the power of 0 is 1, we get the equation: -2 = a.
Substitute this value of a into the other equation: -50 = (-2)b^2.
Solving for b, we have b^2 = 25, which means b = ±5.
Now we have two potential equations for the exponential function: y = -2 * 5^x and y = -2 * (-5)^x.
Let's substitute x = -2 into both equations to find the value of y:
For y = -2 * 5^(-2): y = -2 * (1/5^2) = -2/25.
For y = -2 * (-5)^(-2): y = -2 * (1/(-5)^2) = -2/25.
So, the value of y when x = -2 is -2/25. Your answer is correct.
2. Solve 64^x < 32^(x + 2).
To compare the two exponential expressions, we need to find a common base. Here, both 64 and 32 can be represented as powers of 2 (2^6 and 2^5, respectively).
Rewrite the equation using the common base:
(2^6)^x < (2^5)^(x + 2).
Next, apply the power of a power rule:
2^(6x) < 2^(5x + 10).
Since the bases are the same, we can drop them and compare the exponents:
6x < 5x + 10.
Simplifying the inequality: 6x - 5x < 10, which gives x < 10.
Thus, the answer is x < 10. Your answer is correct.
3. Write the equation log(243)81 = 4/5.
To solve this equation, we need to rewrite it using exponential form.
The equation log(base, argument)exponent = value can be written as base^(value) = argument^(exponent).
Applying this to the given equation, we get: 243^(4/5) = 81.
So, the equation is 243^(4/5) = 81. Your answer is correct.
4. Evaluate 9^(log(954)).
To evaluate this expression, we need to use the property: a^(log(base)a) = base.
In this case, the base is 9 and the argument of the logarithm is 954.
So, 9^(log(954)) simplifies to 954.
Thus, the answer is 954. Your answer is correct.
5. Solve log(1/8)x = -1.
To solve this logarithmic equation, we need to rewrite it in exponential form.
The logarithmic equation log(base)x = y is equivalent to the exponential equation base^y = x.
Applying this to the given equation, we get: (1/8)^(-1) = x.
Recall that any number (except zero) raised to the power of -1 is equal to its reciprocal.
So, (1/8)^(-1) = 8/1 = 8.
Thus, the solution is x = 8. Your answer is correct.
6. Solve log(base 2)(7x - 3) ≥ log(base 2)(x + 12).
To solve this logarithmic inequality, we need to compare the expressions by using properties of logarithms.
The inequality log(base a)b ≥ log(base a)c is equivalent to b ≥ c.
Applying this to the given inequality, we have: 7x - 3 ≥ x + 12.
Solving for x: 7x - x ≥ 12 + 3.
Simplifying: 6x ≥ 15.
Dividing both sides of the inequality by 6: x ≥ 15/6.
Thus, the solution is x ≥ 5/2. Your answer is correct.
Overall, all of your answers are correct. Well done!