What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places.$\[ 4 + 4 \cdot \log_2 X = 14 \\]A. \[$ X = 6.25 \$\] B. \[$ X = 3.06 \$\] C. \[$ X = 3.36 \$\] D. \[$ X = 5.66 \$\]

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Solving the Equation: 4 + 4 * log2(x) = 14

In this article, we will be solving the equation 4 + 4 * log2(x) = 14. This equation involves logarithms and requires us to isolate the variable x. We will use algebraic manipulations and properties of logarithms to solve for x.

The given equation is 4 + 4 * log2(x) = 14. This equation involves a logarithmic term, log2(x), which is the logarithm of x to the base 2. The logarithm of a number is the exponent to which the base must be raised to produce that number. In this case, the base is 2.

To solve for x, we need to isolate the logarithmic term. We can start by subtracting 4 from both sides of the equation:

4 + 4 * log2(x) - 4 = 14 - 4

This simplifies to:

4 * log2(x) = 10

Next, we can divide both sides of the equation by 4:

(4 * log2(x)) / 4 = 10 / 4

This simplifies to:

log2(x) = 2.5

To solve for x, we need to get rid of the logarithm. We can do this by exponentiating both sides of the equation. Since the base of the logarithm is 2, we can use the fact that 2^y = x if and only if log2(x) = y. Therefore, we can exponentiate both sides of the equation:

2^log2(x) = 2^2.5

This simplifies to:

x = 2^2.5

To evaluate the exponent, we can use the fact that 2^y = e^(y * ln(2)), where e is the base of the natural logarithm and ln is the natural logarithm. Therefore, we can rewrite the exponent as:

x = e^(2.5 * ln(2))

Using a calculator, we can evaluate the exponent:

x ≈ 5.66

In this article, we solved the equation 4 + 4 * log2(x) = 14. We isolated the logarithmic term, divided by 4, and exponentiated both sides to solve for x. The solution is x ≈ 5.66.

The correct answer is D. x = 5.66.

This problem involves logarithms and requires us to use algebraic manipulations and properties of logarithms to solve for x. The solution involves isolating the logarithmic term, dividing by 4, and exponentiating both sides. The final answer is x ≈ 5.66.

  • Logarithmic equations
  • Algebraic manipulations
  • Properties of logarithms
  • Exponentiation
  • [1] "Logarithmic Equations" by Math Open Reference
  • [2] "Algebraic Manipulations" by Khan Academy
  • [3] "Properties of Logarithms" by Wolfram MathWorld
  • [4] "Exponentiation" by Math Is Fun
    Solving the Equation: 4 + 4 * log2(x) = 14 - Q&A

In our previous article, we solved the equation 4 + 4 * log2(x) = 14. We isolated the logarithmic term, divided by 4, and exponentiated both sides to solve for x. The solution was x ≈ 5.66. In this article, we will answer some frequently asked questions about the solution and provide additional insights.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is 2. This means that the logarithm is a logarithm to the base 2.

Q: How do I isolate the logarithmic term in the equation?

A: To isolate the logarithmic term, you can subtract 4 from both sides of the equation. This will give you 4 * log2(x) = 10.

Q: Why do I need to divide by 4?

A: You need to divide by 4 because the equation is 4 * log2(x) = 10. Dividing by 4 will give you log2(x) = 2.5.

Q: What is the relationship between the logarithm and the exponent?

A: The logarithm and the exponent are related by the fact that 2^y = x if and only if log2(x) = y. This means that if you exponentiate both sides of the equation, you will get x = 2^2.5.

Q: How do I evaluate the exponent?

A: To evaluate the exponent, you can use the fact that 2^y = e^(y * ln(2)), where e is the base of the natural logarithm and ln is the natural logarithm. You can then use a calculator to evaluate the exponent.

Q: What is the solution to the equation?

A: The solution to the equation is x ≈ 5.66.

Q: Why is the solution not one of the other options?

A: The solution is not one of the other options because the other options are not correct. The other options are x = 6.25, x = 3.06, and x = 3.36. However, these values do not satisfy the equation.

Q: What are some related topics to this problem?

A: Some related topics to this problem include logarithmic equations, algebraic manipulations, properties of logarithms, and exponentiation.

Q: Where can I find more information about logarithmic equations?

A: You can find more information about logarithmic equations on websites such as Math Open Reference, Khan Academy, Wolfram MathWorld, and Math Is Fun.

In this article, we answered some frequently asked questions about the solution to the equation 4 + 4 * log2(x) = 14. We provided additional insights and related topics to help you better understand the solution.

  • Logarithmic equations
  • Algebraic manipulations
  • Properties of logarithms
  • Exponentiation
  • [1] "Logarithmic Equations" by Math Open Reference
  • [2] "Algebraic Manipulations" by Khan Academy
  • [3] "Properties of Logarithms" by Wolfram MathWorld
  • [4] "Exponentiation" by Math Is Fun