Evaluate The Expression. Simplify Your Answer Completely.(a) $\log_8(64$\] $\square$(b) $\log_7(49$\] $\square$(c) $\log_9\left(9^{13}\right$\] $\square$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will evaluate three logarithmic expressions and simplify our answers completely. We will use the properties of logarithms to solve these equations and provide a step-by-step explanation of each solution.

Evaluating Logarithmic Expressions

(a) $\log_8(64)$

To evaluate this expression, we need to find the power to which the base (8) must be raised to obtain the number inside the logarithm (64). In other words, we need to find the exponent that satisfies the equation $8^x = 64$.

We can rewrite 64 as $8^2$, since $8^2 = 64$. Therefore, we can conclude that $\log_8(64) = 2$.

(b) $\log_7(49)$

To evaluate this expression, we need to find the power to which the base (7) must be raised to obtain the number inside the logarithm (49). In other words, we need to find the exponent that satisfies the equation $7^x = 49$.

We can rewrite 49 as $7^2$, since $7^2 = 49$. Therefore, we can conclude that $\log_7(49) = 2$.

(c) $\log_9\left(9^{13}\right)$

To evaluate this expression, we need to find the power to which the base (9) must be raised to obtain the number inside the logarithm, which is already in the form of a power of 9. In other words, we need to find the exponent that satisfies the equation $9^x = 9^{13}$.

Since the bases are the same, we can equate the exponents, which gives us $x = 13$. Therefore, we can conclude that $\log_9\left(9^{13}\right) = 13$.

Properties of Logarithms

Logarithmic equations can be simplified using the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y\log_b(x)$

These properties can be used to simplify logarithmic expressions and solve equations.

Example 1: Simplifying a Logarithmic Expression

Let's consider the expression $\log_2(16)$. We can rewrite 16 as $2^4$, since $2^4 = 16$. Using the power property of logarithms, we can simplify the expression as follows:

$\log_2(16) = \log_2(2^4) = 4\log_2(2) = 4$

Therefore, the simplified expression is $\log_2(16) = 4$.

Example 2: Solving a Logarithmic Equation

Let's consider the equation $\log_3(x) = 2$. We can rewrite the equation as $3^2 = x$, since $\log_3(x) = 2$ implies that $x = 3^2$. Therefore, the solution to the equation is $x = 9$.

Conclusion

In this article, we evaluated three logarithmic expressions and simplified our answers completely. We used the properties of logarithms to solve these equations and provide a step-by-step explanation of each solution. We also discussed the three main properties of logarithms and provided examples of how to simplify logarithmic expressions and solve equations.

Final Thoughts

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. By understanding the properties of logarithms and how to simplify logarithmic expressions, we can solve equations and make predictions about real-world phenomena.

References

• [1] "Logarithms" by Khan Academy
• [2] "Properties of Logarithms" by Math Is Fun
• [3] "Logarithmic Equations" by Purplemath

Further Reading

• [1] "Introduction to Logarithms" by MIT OpenCourseWare
• [2] "Logarithmic Functions" by Wolfram MathWorld
• [3] "Logarithmic Equations and Inequalities" by Mathway

Note: The references and further reading sections are not included in the word count.

Introduction

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that involves a power to which a base must be raised to obtain a given number.

Q: What are the properties of logarithms?

A: The three main properties of logarithms are:

• Product Property: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y\log_b(x)$

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms. For example, if you have the expression $\log_2(16)$, you can rewrite 16 as $2^4$, since $2^4 = 16$. Using the power property of logarithms, you can simplify the expression as follows:

$\log_2(16) = \log_2(2^4) = 4\log_2(2) = 4$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms. For example, if you have the equation $\log_3(x) = 2$, you can rewrite the equation as $3^2 = x$, since $\log_3(x) = 2$ implies that $x = 3^2$. Therefore, the solution to the equation is $x = 9$.

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 $\log_2(x) = 3$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a graphing software. You can also use the properties of logarithms to graph the function. For example, if you have the function $y = \log_2(x)$, you can graph the function by plotting the points $(1, 0)$, $(2, 1)$, $(4, 2)$, and so on.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in various fields, including physics, engineering, and computer science. Some common applications include:

• Sound levels: Logarithmic equations are used to measure sound levels in decibels.
• Light intensity: Logarithmic equations are used to measure light intensity in lux.
• Chemical reactions: Logarithmic equations are used to model chemical reactions and predict the rate of reaction.
• Population growth: Logarithmic equations are used to model population growth and predict the future population size.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Forgetting to change the base: When solving a logarithmic equation, make sure to change the base of the logarithm to the same base as the equation.
• Not using the properties of logarithms: Make sure to use the properties of logarithms to simplify the equation and solve for the variable.
• Not checking the domain: Make sure to check the domain of the logarithmic function to ensure that the solution is valid.

Conclusion

In this article, we provided a Q&A guide to help you understand logarithmic equations and how to solve them. We covered topics such as the properties of logarithms, simplifying logarithmic expressions, solving logarithmic equations, and graphing logarithmic functions. We also discussed some common applications of logarithmic equations and some common mistakes to avoid when solving them.

Final Thoughts

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. By understanding logarithmic equations and how to solve them, you can make predictions about real-world phenomena and solve problems in a variety of fields.

References

• [1] "Logarithms" by Khan Academy
• [2] "Properties of Logarithms" by Math Is Fun
• [3] "Logarithmic Equations" by Purplemath

Further Reading

• [1] "Introduction to Logarithms" by MIT OpenCourseWare
• [2] "Logarithmic Functions" by Wolfram MathWorld
• [3] "Logarithmic Equations and Inequalities" by Mathway