Which Of These Is The First Step For Solving The Equation Below? Log ⁡ 6 ( X + 9 ) = 28 \log _6(x+9)=28 Lo G 6 ​ ( X + 9 ) = 28 A. Divide Both Sides By 28. B. Take The Log ⁡ 6 \log _6 Lo G 6 ​ Of Both Sides. C. Use The Definition Of Logarithms To Convert Log ⁡ 6 ( X + 9 ) = 28 \log _6(x+9)=28 Lo G 6 ​ ( X + 9 ) = 28 To An

Understanding Logarithmic Equations

Logarithmic equations are a type of mathematical equation that involves logarithms. A logarithm is the inverse operation of exponentiation, and it is used to solve equations that involve exponential expressions. In this article, we will focus on solving logarithmic equations of the form $\log _a(x+b)=c$, where $a$ is the base of the logarithm, $x$ is the variable, $b$ is a constant, and $c$ is the result of the logarithm.

The Given Equation

The given equation is $\log _6(x+9)=28$. This equation involves a logarithm with base 6, and we need to solve for the variable $x$. To do this, we need to follow a series of steps that will help us isolate the variable and find its value.

Step 1: Understanding the Definition of Logarithms

Before we start solving the equation, it's essential to understand the definition of logarithms. The logarithm $\log _a(x)$ is defined as the exponent to which the base $a$ must be raised to produce the number $x$. In other words, if $\log _a(x)=y$, then $a^y=x$. This definition is crucial in solving logarithmic equations.

Step 2: Identifying the First Step

Now that we have a clear understanding of logarithmic equations and the definition of logarithms, let's identify the first step in solving the given equation. The first step is to use the definition of logarithms to convert $\log _6(x+9)=28$ to an exponential equation.

Using the Definition of Logarithms

Using the definition of logarithms, we can rewrite the given equation as $6^{28}=x+9$. This is an exponential equation, and we can solve it by isolating the variable $x$.

Step 3: Isolating the Variable

To isolate the variable $x$, we need to subtract 9 from both sides of the equation. This will give us $x=6^{28}-9$.

Step 4: Evaluating the Expression

Now that we have isolated the variable $x$, we can evaluate the expression $6^{28}-9$. This will give us the value of $x$.

Conclusion

In conclusion, the first step in solving the equation $\log _6(x+9)=28$ is to use the definition of logarithms to convert the equation to an exponential equation. This involves rewriting the equation as $6^{28}=x+9$. From there, we can isolate the variable $x$ and evaluate the expression to find its value.

Answer

The correct answer is C. Use the definition of logarithms to convert $\log _6(x+9)=28$ to an exponential equation.

Additional Tips and Tricks

• When solving logarithmic equations, it's essential to understand the definition of logarithms and how to use it to convert the equation to an exponential equation.
• Always isolate the variable $x$ before evaluating the expression.
• Use the properties of logarithms, such as the product rule and the quotient rule, to simplify the equation and make it easier to solve.

Common Mistakes to Avoid

• Not understanding the definition of logarithms and how to use it to convert the equation to an exponential equation.
• Not isolating the variable $x$ before evaluating the expression.
• Not using the properties of logarithms to simplify the equation and make it easier to solve.

Real-World Applications

Logarithmic equations have many real-world applications, such as:

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Solving problems in physics and engineering

Conclusion

Frequently Asked Questions

Q: What is the first step in solving a logarithmic equation? A: The first step in solving a logarithmic equation is to use the definition of logarithms to convert the equation to an exponential equation.

Q: How do I convert a logarithmic equation to an exponential equation? A: To convert a logarithmic equation to an exponential equation, you need to use the definition of logarithms. For example, if the equation is $\log _a(x+b)=c$, you can rewrite it as $a^c=x+b$.

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 $\log _a(x+b)=c$ is a logarithmic equation, while the equation $a^c=x+b$ is an exponential equation.

Q: How do I solve an exponential equation? A: To solve an exponential equation, you need to isolate the variable $x$. This can be done by using algebraic operations, such as addition, subtraction, multiplication, and division.

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 definition of logarithms and how to use it to convert the equation to an exponential equation.
• Not isolating the variable $x$ before evaluating the expression.
• Not using the properties of logarithms to simplify the equation and make it easier to solve.

Q: What are some real-world applications of logarithmic equations? A: Logarithmic equations have many real-world applications, such as:

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Solving problems in physics and engineering

Q: How do I evaluate an expression that involves a logarithm? A: To evaluate an expression that involves a logarithm, you need to use the definition of logarithms. For example, if the expression is $\log _a(x+b)$, you can rewrite it as $a^{\log _a(x+b)}=x+b$.

Q: What is the product rule of logarithms? A: The product rule of logarithms states that $\log _a(xy)=\log _a(x)+\log _a(y)$. This rule can be used to simplify logarithmic expressions and make them easier to solve.

Q: What is the quotient rule of logarithms? A: The quotient rule of logarithms states that $\log _a(\frac{x}{y})=\log _a(x)-\log _a(y)$. This rule can be used to simplify logarithmic expressions and make them easier to solve.

Q: How do I use the product rule and quotient rule of logarithms to simplify a logarithmic expression? A: To use the product rule and quotient rule of logarithms to simplify a logarithmic expression, you need to apply the rules to the expression and simplify it. For example, if the expression is $\log _a(xy)$, you can use the product rule to rewrite it as $\log _a(x)+\log _a(y)$.

Conclusion

In conclusion, logarithmic equations are a type of mathematical equation that involves logarithms. By understanding the definition of logarithms and how to use it to convert the equation to an exponential equation, you can solve logarithmic equations and apply them to real-world problems. This article has provided a comprehensive guide to logarithmic equations, including frequently asked questions and answers.