Solve For { X $}$: { 2^ X} = 16 $}$ (1 Mark)11. Express { 4^2 = 16 $}$ In Logarithmic Form (1 Mark)12. Express { \log_5 25 = 2 $ $ In Exponential Form: (1 Mark)13. If { \log_{10} X = 3 $}$,

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Introduction

Exponential equations and logarithmic functions are fundamental concepts in mathematics, particularly in algebra and calculus. In this article, we will explore how to solve exponential equations and convert between logarithmic and exponential forms. We will also provide examples and explanations to help you understand these concepts.

Solving Exponential Equations

Exponential equations are equations that involve exponential functions, which are functions of the form f(x) = a^x, where a is a positive real number and x is the variable. To solve an exponential equation, we need to isolate the variable x.

Example 1: Solving for x in 2^x = 16

To solve for x in the equation 2^x = 16, we can use the fact that 16 = 2^4. Therefore, we can rewrite the equation as:

2^x = 2^4

Since the bases are the same, we can equate the exponents:

x = 4

Therefore, the solution to the equation 2^x = 16 is x = 4.

Example 2: Solving for x in 4^2 = 16

To solve for x in the equation 4^2 = 16, we can use the fact that 4^2 = (22)2 = 2^4. Therefore, we can rewrite the equation as:

2^4 = 16

Since the bases are the same, we can equate the exponents:

x = 4

Therefore, the solution to the equation 4^2 = 16 is x = 4.

Example 3: Solving for x in log_5 25 = 2

To solve for x in the equation log_5 25 = 2, we can use the fact that log_a b = c is equivalent to a^c = b. Therefore, we can rewrite the equation as:

5^2 = 25

Since the bases are the same, we can equate the exponents:

x = 2

Therefore, the solution to the equation log_5 25 = 2 is x = 2.

Converting Between Logarithmic and Exponential Forms

Logarithmic functions and exponential functions are inverse functions, which means that they are related by a simple transformation. To convert between logarithmic and exponential forms, we can use the following formulas:

  • log_a b = c is equivalent to a^c = b
  • a^x = b is equivalent to log_a b = x

Example 1: Converting log_10 x = 3 to exponential form

To convert the equation log_10 x = 3 to exponential form, we can use the formula a^x = b. Therefore, we can rewrite the equation as:

10^3 = x

Therefore, the exponential form of the equation log_10 x = 3 is 10^3 = x.

Example 2: Converting 4^2 = 16 to logarithmic form

To convert the equation 4^2 = 16 to logarithmic form, we can use the formula log_a b = c. Therefore, we can rewrite the equation as:

log_4 16 = 2

Therefore, the logarithmic form of the equation 4^2 = 16 is log_4 16 = 2.

Conclusion

In this article, we have explored how to solve exponential equations and convert between logarithmic and exponential forms. We have provided examples and explanations to help you understand these concepts. By following the formulas and techniques presented in this article, you should be able to solve exponential equations and convert between logarithmic and exponential forms with ease.

Practice Problems

  1. Solve for x in 3^x = 27
  2. Express 9^2 = 81 in logarithmic form
  3. Express log_2 8 = 3 in exponential form

Answers

  1. x = 3
  2. log_9 81 = 2
  3. 2^3 = 8
    Q&A: Exponential Equations and Logarithmic Functions =====================================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponential equations and logarithmic functions.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form f(x) = a^x, where a is a positive real number and x is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable x. You can do this by using the fact that a^x = b is equivalent to log_a b = x.

Q: What is a logarithmic function?

A: A logarithmic function is the inverse of an exponential function. It is a function of the form f(x) = log_a x, where a is a positive real number and x is the variable.

Q: How do I convert between logarithmic and exponential forms?

A: You can convert between logarithmic and exponential forms using the following formulas:

  • log_a b = c is equivalent to a^c = b
  • a^x = b is equivalent to log_a b = x

Q: What is the difference between a logarithmic function and an exponential function?

A: A logarithmic function and an exponential function are inverse functions. This means that they are related by a simple transformation. For example, the logarithmic function log_a x is the inverse of the exponential function a^x.

Q: How do I evaluate a logarithmic function?

A: To evaluate a logarithmic function, you need to find the value of the variable x that satisfies the equation log_a x = c. You can do this by using the fact that log_a x = c is equivalent to a^c = x.

Q: What is the base of a logarithmic function?

A: The base of a logarithmic function is the positive real number a that is used to define the function. For example, in the function log_2 x, the base is 2.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to plot the points (x, log_a x) for a range of values of x. You can use a graphing calculator or a computer program to help you with this.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

  • log_2 x
  • log_10 x
  • log_e x

Q: How do I use logarithmic functions in real-world applications?

A: Logarithmic functions are used in a wide range of real-world applications, including:

  • Finance: Logarithmic functions are used to calculate interest rates and investment returns.
  • Science: Logarithmic functions are used to model population growth and decay.
  • Engineering: Logarithmic functions are used to design and optimize systems.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponential equations and logarithmic functions. We hope that this article has been helpful in clarifying these concepts and providing you with a better understanding of how to use them in real-world applications.

Practice Problems

  1. Solve for x in 2^x = 8
  2. Express 3^2 = 9 in logarithmic form
  3. Express log_5 25 = 2 in exponential form

Answers

  1. x = 3
  2. log_3 9 = 2
  3. 5^2 = 25