What Is The Solution To The Equation Below? Round Your Answer To Two Decimal Places.$\[ \log_2 X = 2.5 \\]A. \[$x = 5.00\$\] B. \[$x = 12.18\$\] C. \[$x = 6.25\$\] D. \[$x = 5.66\$\]
Understanding the Problem
The given equation is a logarithmic equation in the form of log2(x) = 2.5. This equation can be solved by converting it into exponential form, which will allow us to find the value of x.
Converting Logarithmic to Exponential Form
To convert the logarithmic equation log2(x) = 2.5 into exponential form, we need to use the property of logarithms that states: loga(x) = y is equivalent to a^y = x.
Applying the Property
Using the property mentioned above, we can rewrite the equation log2(x) = 2.5 as 2^2.5 = x.
Evaluating the Exponential Expression
Now, we need to evaluate the exponential expression 2^2.5. This can be done by using the fact that 2^2 = 4 and 2^0.5 = √2.
Simplifying the Expression
Using the fact that 2^2 = 4, we can rewrite 2^2.5 as 2^2 * 2^0.5. This simplifies to 4 * √2.
Evaluating the Expression
Now, we need to evaluate the expression 4 * √2. This can be done by multiplying 4 by the square root of 2.
Calculating the Value
The square root of 2 is approximately 1.414. Therefore, 4 * √2 is equal to 4 * 1.414, which is approximately 5.656.
Rounding the Answer
The problem asks us to round our answer to two decimal places. Therefore, we need to round 5.656 to two decimal places.
Final Answer
Rounding 5.656 to two decimal places gives us 5.66.
Conclusion
The solution to the equation log2(x) = 2.5 is x = 5.66.
Comparison with Options
Comparing our answer with the options given, we can see that our answer matches option D.
Final Thoughts
In this problem, we used the property of logarithms to convert the equation into exponential form. We then evaluated the exponential expression and rounded our answer to two decimal places. This problem demonstrates the importance of understanding the properties of logarithms and how to apply them to solve equations.
Key Takeaways
- The equation log2(x) = 2.5 can be solved by converting it into exponential form.
- The exponential form of the equation is 2^2.5 = x.
- The value of 2^2.5 can be evaluated by using the fact that 2^2 = 4 and 2^0.5 = √2.
- The final answer is x = 5.66.
Frequently Asked Questions
- What is the solution to the equation log2(x) = 2.5?
- How do we convert a logarithmic equation into exponential form?
- What is the value of 2^2.5?
- How do we round our answer to two decimal places?
Answers to Frequently Asked Questions
- The solution to the equation log2(x) = 2.5 is x = 5.66.
- A logarithmic equation can be converted into exponential form by using the property of logarithms that states: loga(x) = y is equivalent to a^y = x.
- The value of 2^2.5 is approximately 5.656.
- To round our answer to two decimal places, we need to look at the third decimal place and decide whether to round up or down.
Understanding Logarithmic Equations
Logarithmic equations are a type of mathematical equation that involves logarithms. They are used to solve problems that involve exponential growth or decay. In this article, we will answer some frequently asked questions about logarithmic equations.
Q: What is a logarithmic equation?
A: A logarithmic equation is an equation that involves a logarithm. It is a mathematical equation that can be solved by converting it into exponential form.
Q: How do I convert a logarithmic equation into exponential form?
A: To convert a logarithmic equation into exponential form, you need to use the property of logarithms that states: loga(x) = y is equivalent to a^y = x.
Q: What is the difference between a logarithmic equation and an exponential equation?
A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation log2(x) = 2.5 is a logarithmic equation, while the equation 2^2.5 = x is an exponential equation.
Q: How do I solve a logarithmic equation?
A: To solve a logarithmic equation, you need to convert it into exponential form and then evaluate the exponential expression.
Q: What is the base of a logarithm?
A: The base of a logarithm is the number that is used to raise to a power in the logarithmic equation. For example, in the equation log2(x) = 2.5, the base is 2.
Q: What is the logarithm of a number?
A: The logarithm of a number is the exponent to which the base must be raised to produce the number. For example, the logarithm of 8 to the base 2 is 3, because 2^3 = 8.
Q: How do I evaluate a logarithmic expression?
A: To evaluate a logarithmic expression, you need to use the properties of logarithms, such as the product rule and the quotient rule.
Q: What is the product rule of logarithms?
A: The product rule of logarithms states that loga(x * y) = loga(x) + loga(y).
Q: What is the quotient rule of logarithms?
A: The quotient rule of logarithms states that loga(x / y) = loga(x) - loga(y).
Q: How do I use the product rule and the quotient rule to evaluate a logarithmic expression?
A: To use the product rule and the quotient rule, you need to apply the rules to the logarithmic expression and then simplify the expression.
Q: What is the change of base formula?
A: The change of base formula is a formula that allows you to change the base of a logarithm. It is given by: loga(x) = (logb(x)) / (logb(a)).
Q: How do I use the change of base formula?
A: To use the change of base formula, you need to apply the formula to the logarithmic expression and then simplify the expression.
Q: What are some common logarithmic equations?
A: Some common logarithmic equations include log2(x) = 2.5, log10(x) = 1.5, and loge(x) = 2.
Q: How do I solve a logarithmic equation with a negative exponent?
A: To solve a logarithmic equation with a negative exponent, you need to use the property of logarithms that states: loga(x) = -y is equivalent to a^(-y) = x.
Q: What is the relationship between logarithmic equations and exponential equations?
A: Logarithmic equations and exponential equations are related by the property of logarithms that states: loga(x) = y is equivalent to a^y = x.
Q: How do I use logarithmic equations to solve real-world problems?
A: Logarithmic equations can be used to solve real-world problems that involve exponential growth or decay. For example, you can use logarithmic equations to model population growth or to calculate the half-life of a radioactive substance.
Q: What are some common applications of logarithmic equations?
A: Some common applications of logarithmic equations include finance, science, and engineering.
Q: How do I use logarithmic equations to solve problems in finance?
A: Logarithmic equations can be used to solve problems in finance that involve compound interest or exponential growth.
Q: How do I use logarithmic equations to solve problems in science?
A: Logarithmic equations can be used to solve problems in science that involve exponential growth or decay.
Q: How do I use logarithmic equations to solve problems in engineering?
A: Logarithmic equations can be used to solve problems in engineering that involve exponential growth or decay.
Conclusion
In this article, we have answered some frequently asked questions about logarithmic equations. We have discussed the properties of logarithms, how to convert logarithmic equations into exponential form, and how to solve logarithmic equations. We have also discussed some common applications of logarithmic equations and how to use them to solve real-world problems.