\[$-7 \cdot 3^{0.25z} = -10\$\]Which Of The Following Is The Solution Of The Equation? Choose 1 Answer:A. \[$x = 4 \log_{\frac{10}{7}}(3)\$\]B. \[$x = \frac{4 \log_3(7)}{\log_3 10}\$\]C. \[$x = \frac{4

Solving the Equation: $-7 \cdot 3^{0.25z} = -10$

In this article, we will be solving the equation $-7 \cdot 3^{0.25z} = -10$. This equation involves a logarithmic function and requires us to manipulate the equation to isolate the variable $z$. We will use various mathematical techniques, including logarithmic properties and exponent rules, to solve for $z$.

Step 1: Isolate the Exponential Term

The first step in solving this equation is to isolate the exponential term. We can do this by dividing both sides of the equation by $-7$.

$$-7 \cdot 3^{0.25z} = -10$$

$$3^{0.25z} = \frac{-10}{-7}$$

$$3^{0.25z} = \frac{10}{7}$$

Step 2: Use Logarithmic Properties

Next, we can use logarithmic properties to solve for $z$. We can take the logarithm of both sides of the equation, using the base $3$.

$$\log_3 3^{0.25z} = \log_3 \frac{10}{7}$$

Using the property of logarithms that states $\log_b b^x = x$, we can simplify the left-hand side of the equation.

$$0.25z = \log_3 \frac{10}{7}$$

Step 3: Solve for $z$

Now that we have isolated $z$, we can solve for its value. We can do this by dividing both sides of the equation by $0.25$.

$$z = \frac{\log_3 \frac{10}{7}}{0.25}$$

Using the property of logarithms that states $\log_b \frac{a}{c} = \log_b a - \log_b c$, we can simplify the numerator of the right-hand side.

$$z = \frac{\log_3 10 - \log_3 7}{0.25}$$

Step 4: Simplify the Expression

Finally, we can simplify the expression by using the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$.

$$z = \frac{\frac{\log 10}{\log 3} - \frac{\log 7}{\log 3}}{0.25}$$

$$z = \frac{\frac{\log 10}{\log 3} - \frac{\log 7}{\log 3}}{\frac{1}{4}}$$

$$z = 4 \left(\frac{\log 10}{\log 3} - \frac{\log 7}{\log 3}\right)$$

$$z = 4 \log_3 \frac{10}{7}$$

In conclusion, the solution to the equation $-7 \cdot 3^{0.25z} = -10$ is $x = 4 \log_3 \frac{10}{7}$. This solution was obtained by isolating the exponential term, using logarithmic properties, and simplifying the expression.

Let's compare our solution with the other options provided.

• Option A: $x = 4 \log_{\frac{10}{7}}(3)$

  This option is incorrect because the base of the logarithm is $\frac{10}{7}$, whereas our solution has a base of $3$.

• Option B: $x = \frac{4 \log_3(7)}{\log_3 10}$

  This option is also incorrect because the numerator and denominator have the wrong signs.

• Option C: $x = \frac{4 \log_3(10)}{\log_3 7}$

  This option is incorrect because the numerator and denominator have the wrong signs.

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In our previous article, we solved the equation $-7 \cdot 3^{0.25z} = -10$ and obtained the solution $x = 4 \log_3 \frac{10}{7}$. In this article, we will answer some frequently asked questions about the solution and the equation.

Q: What is the base of the logarithm in the solution?

A: The base of the logarithm in the solution is $3$. This is because we used the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$ to simplify the expression.

Q: Why did we use the property of logarithms to simplify the expression?

A: We used the property of logarithms to simplify the expression because it allowed us to rewrite the logarithm in terms of a common base. This made it easier to solve for $z$.

Q: What is the relationship between the logarithm and the exponential function?

A: The logarithm and the exponential function are inverse functions. This means that if $y = b^x$, then $x = \log_b y$. In our solution, we used this relationship to solve for $z$.

Q: How did we isolate the exponential term in the equation?

A: We isolated the exponential term by dividing both sides of the equation by $-7$. This allowed us to rewrite the equation as $3^{0.25z} = \frac{10}{7}$.

Q: What is the significance of the coefficient $0.25$ in the equation?

A: The coefficient $0.25$ is significant because it represents the exponent of the base $3$. This means that the base $3$ is raised to the power of $0.25z$.

Q: How did we solve for $z$?

A: We solved for $z$ by using the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$. We then simplified the expression to obtain the solution $x = 4 \log_3 \frac{10}{7}$.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 4 \log_3 \frac{10}{7}$.

In conclusion, we have answered some frequently asked questions about the solution and the equation $-7 \cdot 3^{0.25z} = -10$. We hope that this Q&A article has been helpful in clarifying any doubts that you may have had.

If you would like to learn more about logarithms and exponential functions, we recommend the following resources:

• Khan Academy: Logarithms and Exponential Functions
• Mathway: Logarithms and Exponential Functions
• Wolfram Alpha: Logarithms and Exponential Functions

We hope that you find these resources helpful in your studies.