Given That The Point \[$(8,3)\$\] Lies On The Graph Of \[$g(x)=\log _2 X\$\], Which Point Lies On The Graph Of \[$f(x)=\log _2(x+3)+2\$\]?A. \[$(5,1)\$\]B. \[$(5,5)\$\]C. \[$(11,1)\$\]D. \[$(11,5)\$\]

by ADMIN 201 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will explore how to solve logarithmic equations, with a focus on the given problem involving the functions {g(x)=\log _2 x$}$ and {f(x)=\log _2(x+3)+2$}$.

Understanding Logarithmic Functions

Before we dive into the problem, let's take a moment to understand the concept of logarithmic functions. A logarithmic function is the inverse of an exponential function. In other words, if {y = 2^x$}$, then {x = \log_2 y$}$. The logarithmic function {\log_b x$}$ gives us the power to which the base {b$}$ must be raised to obtain the number {x$}$.

The Given Problem

The problem states that the point {(8,3)$}$ lies on the graph of {g(x)=\log _2 x$}$. We are asked to find the point that lies on the graph of {f(x)=\log _2(x+3)+2$}$. To solve this problem, we need to understand the relationship between the two functions.

Substitution Method

One way to solve this problem is to use the substitution method. We can substitute {x = 8$}$ into the function {f(x)$}$ to find the corresponding value of {y$}$.

{f(8) = \log_2(8+3) + 2$}$

{f(8) = \log_2(11) + 2$}$

{f(8) = 3.46 + 2$}$

{f(8) = 5.46$}$

However, we are not given the exact value of {f(8)$}$, but rather a set of possible points that lie on the graph of {f(x)$}$. We need to find the point that corresponds to the value {f(8) = 5.46$}$.

Analyzing the Options

Let's analyze the options given:

A. {(5,1)$}$ B. {(5,5)$}$ C. {(11,1)$}$ D. {(11,5)$}$

We can see that option B, {(5,5)$}$, is the only option that has a {y$}$-value of ${5\$}, which is close to the value we obtained for {f(8)$}$.

Conclusion

Based on our analysis, we can conclude that the point that lies on the graph of {f(x)=\log _2(x+3)+2$}$ is option B, {(5,5)$}$.

Final Answer

The final answer is option B, {(5,5)$}$.

Additional Tips and Tricks

Here are some additional tips and tricks to help you solve logarithmic equations:

  • Make sure to understand the concept of logarithmic functions and their properties.
  • Use the substitution method to solve logarithmic equations.
  • Analyze the options given and look for the one that matches the value you obtained.
  • Practice solving logarithmic equations to become more comfortable with the concept.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving logarithmic equations:

  • Not understanding the concept of logarithmic functions and their properties.
  • Not using the substitution method to solve logarithmic equations.
  • Not analyzing the options given and looking for the one that matches the value obtained.
  • Not practicing solving logarithmic equations to become more comfortable with the concept.

Conclusion

Q: What is a logarithmic function?

A: A logarithmic function is the inverse of an exponential function. In other words, if {y = 2^x$}$, then {x = \log_2 y$}$. The logarithmic function {\log_b x$}$ gives us the power to which the base {b$}$ must be raised to obtain the number {x$}$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the substitution method. This involves substituting the value of the variable into the equation and solving for the other variable. For example, if we have the equation {\log_2 x = 3$}$, we can substitute {x = 2^3$}$ to solve for {x$}$.

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

A: A logarithmic function is the inverse of an exponential function. In other words, if {y = 2^x$}$, then {x = \log_2 y$}$. This means that logarithmic functions and exponential functions are related, but they are not the same thing.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the variable that makes the expression true. For example, if we have the expression {\log_2 (x+3) = 2$}$, we need to find the value of {x$}$ that makes the expression true.

Q: What is the domain and range of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or graph paper. You can also use the properties of logarithmic functions to graph them. For example, the graph of {\log_2 x$}$ is a curve that increases as {x$}$ increases.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

  • {\log_2 x$}$
  • {\log_3 x$}$
  • {\log_5 x$}$
  • {\log_{10} x$}$

Q: How do I solve a logarithmic equation with a base other than 10?

A: To solve a logarithmic equation with a base other than 10, you can use the change of base formula. This formula allows you to change the base of a logarithmic expression to a different base.

Q: What is the change of base formula?

A: The change of base formula is:

{\log_b x = \frac{\log_a x}{\log_a b}$}$

This formula allows you to change the base of a logarithmic expression to a different base.

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to substitute the values of {x$}$, {a$}$, and {b$}$ into the formula. For example, if we have the expression {\log_3 x = 2$}$ and we want to change the base to 10, we can use the change of base formula to get:

{\log_3 x = \frac{\log_{10} x}{\log_{10} 3}$}$

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

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

  • Not understanding the concept of logarithmic functions and their properties.
  • Not using the substitution method to solve logarithmic equations.
  • Not analyzing the options given and looking for the one that matches the value obtained.
  • Not practicing solving logarithmic equations to become more comfortable with the concept.

Conclusion

In conclusion, logarithmic equations are an important concept in mathematics, and understanding how to solve them is crucial for solving various mathematical problems. By using the substitution method and analyzing the options given, we can find the point that lies on the graph of {f(x)=\log _2(x+3)+2$}$. With practice and patience, you can become more comfortable with solving logarithmic equations and apply this knowledge to real-world problems.