The Solution To The Equation $\sigma\left(2^{x+4}\right)=36$ Is:1. -1 2. $\frac{\ln 36}{\ln 12}-4$ 3. $\ln (3)-4$ 4. $\frac{\ln 6}{\ln 2}-4$

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Understanding the Problem

The given equation is $\sigma\left(2^{x+4}\right)=36$, where $\sigma$ represents the logarithmic function. To solve this equation, we need to first understand the properties of logarithms and how to manipulate them to isolate the variable $x$. The logarithmic function $\sigma$ is defined as $\sigma(x) = \log_b(x)$, where $b$ is the base of the logarithm.

Properties of Logarithms

Before we proceed to solve the equation, let's recall some important properties of logarithms:

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Rule: $\log_b(x^y) = y\log_b(x)$
• Change of Base Formula: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

Manipulating the Equation

Now, let's manipulate the given equation to isolate the variable $x$. We can start by applying the power rule of logarithms to the left-hand side of the equation:

$$\sigma\left(2^{x+4}\right) = \log_b(2^{x+4}) = (x+4)\log_b(2)$$

Since the base of the logarithm is not specified, we can assume that it is the natural logarithm, denoted by $\ln$. Therefore, we can rewrite the equation as:

$$(x+4)\ln(2) = \ln(36)$$

Solving for $x$

Now, we can solve for $x$ by isolating it on one side of the equation. We can start by dividing both sides of the equation by $\ln(2)$:

$$x+4 = \frac{\ln(36)}{\ln(2)}$$

Next, we can subtract $4$ from both sides of the equation to isolate $x$:

$$x = \frac{\ln(36)}{\ln(2)} - 4$$

Evaluating the Options

Now, let's evaluate the given options to see which one matches the solution we obtained:

1. -1: This option does not match our solution.
2. $\frac{\ln 36}{\ln 12}-4$: This option is close to our solution, but it is not exactly the same. We can simplify it by applying the quotient rule of logarithms:

$$\frac{\ln 36}{\ln 12} = \frac{\ln 6^2}{\ln 12} = \frac{2\ln 6}{\ln 12} = \frac{\ln 6}{\ln 2}$$

Substituting this expression into the option, we get:

$$\frac{\ln 36}{\ln 12}-4 = \frac{\ln 6}{\ln 2}-4$$

This option matches our solution. 3. $\ln (3)-4$: This option does not match our solution. 4. $\frac{\ln 6}{\ln 2}-4$: This option matches our solution.

Conclusion

In conclusion, the solution to the equation $\sigma\left(2^{x+4}\right)=36$ is $\frac{\ln 6}{\ln 2}-4$. This solution was obtained by manipulating the equation using the properties of logarithms and isolating the variable $x$.

Final Answer

The final answer is $\boxed{\frac{\ln 6}{\ln 2}-4}$.

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Understanding the Problem

The given equation is $\sigma\left(2^{x+4}\right)=36$, where $\sigma$ represents the logarithmic function. To solve this equation, we need to first understand the properties of logarithms and how to manipulate them to isolate the variable $x$. The logarithmic function $\sigma$ is defined as $\sigma(x) = \log_b(x)$, where $b$ is the base of the logarithm.

Q&A: Solving the Equation

Q: What is the first step in solving the equation $\sigma\left(2^{x+4}\right)=36$?

A: The first step is to apply the power rule of logarithms to the left-hand side of the equation. This will allow us to simplify the expression and isolate the variable $x$.

Q: How do we apply the power rule of logarithms to the left-hand side of the equation?

A: We can apply the power rule of logarithms by rewriting the expression as $(x+4)\log_b(2)$.

Q: What is the next step in solving the equation?

A: The next step is to isolate the variable $x$ by dividing both sides of the equation by $\ln(2)$.

Q: Why do we divide both sides of the equation by $\ln(2)$?

A: We divide both sides of the equation by $\ln(2)$ because it is the coefficient of the variable $x$ in the expression $(x+4)\ln(2)$.

Q: What is the final step in solving the equation?

A: The final step is to subtract $4$ from both sides of the equation to isolate the variable $x$.

Q: What is the solution to the equation $\sigma\left(2^{x+4}\right)=36$?

A: The solution to the equation $\sigma\left(2^{x+4}\right)=36$ is $\frac{\ln 6}{\ln 2}-4$.

Common Mistakes to Avoid

Q: What is a common mistake to avoid when solving the equation $\sigma\left(2^{x+4}\right)=36$?

A: A common mistake to avoid is to forget to apply the power rule of logarithms to the left-hand side of the equation.

Q: What is another common mistake to avoid when solving the equation $\sigma\left(2^{x+4}\right)=36$?

A: Another common mistake to avoid is to forget to isolate the variable $x$ by dividing both sides of the equation by $\ln(2)$.

Conclusion

In conclusion, solving the equation $\sigma\left(2^{x+4}\right)=36$ requires a clear understanding of the properties of logarithms and how to manipulate them to isolate the variable $x$. By following the steps outlined in this article, you can solve the equation and find the solution.

Final Answer

The final answer is $\boxed{\frac{\ln 6}{\ln 2}-4}$.

Additional Resources

• For more information on logarithms and how to manipulate them, see the article "Logarithms: A Comprehensive Guide".
• For more practice problems on solving equations with logarithms, see the article "Solving Equations with Logarithms: Practice Problems".

Related Articles

• "Logarithms: A Comprehensive Guide"
• "Solving Equations with Logarithms: Practice Problems"
• "The Properties of Logarithms"