Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation? ${\log _5 X-\log _5 25=7}$A. ${5^9=x}$B. ${7^5=x}$C. ${5^5=x}$D. ${7^9=x}$

Understanding the Basics of Exponential and Logarithmic Equations

Exponential and logarithmic equations are fundamental concepts in mathematics that are used to solve various problems in different fields, including science, engineering, and finance. In this article, we will focus on solving exponential and logarithmic equations, with a specific emphasis on converting logarithmic equations to exponential equations.

The Given Logarithmic Equation

The given logarithmic equation is ${\log _5 x-\log _5 25=7}$. To solve this equation, we need to apply the properties of logarithms. The first step is to simplify the equation by combining the two logarithmic terms.

Applying the Properties of Logarithms

Using the property of logarithms that states ${\log _a b - \log _a c = \log _a \frac{b}{c}}$, we can rewrite the given equation as ${\log _5 \frac{x}{25}=7}$. This simplifies the equation and makes it easier to solve.

Converting the Logarithmic Equation to Exponential Form

To convert the logarithmic equation to exponential form, we need to apply the definition of logarithms. The definition of logarithms states that if ${\log _a b = c}$, then ${a^c = b}$. Using this definition, we can rewrite the equation as ${5^7 = \frac{x}{25}}$.

Solving for x

To solve for x, we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by 25, which gives us ${x = 5^7 \times 25}$.

Evaluating the Exponential Expression

To evaluate the exponential expression, we need to calculate the value of $5^7$. Using a calculator or a table of exponential values, we can find that $5^7 = 78125$. Therefore, the value of x is ${x = 78125 \times 25 = 1953125}$.

Comparing the Solution to the Answer Choices

Now that we have solved the equation, we can compare our solution to the answer choices. The correct answer is ${5^9 = x}$, which is equivalent to our solution.

Conclusion

In this article, we have discussed how to solve exponential and logarithmic equations, with a specific emphasis on converting logarithmic equations to exponential equations. We have applied the properties of logarithms and the definition of logarithms to simplify and solve the given equation. Our solution is equivalent to the answer choice ${5^9 = x}$, which is the correct answer.

Key Takeaways

• Exponential and logarithmic equations are fundamental concepts in mathematics that are used to solve various problems in different fields.
• The properties of logarithms can be used to simplify logarithmic equations.
• The definition of logarithms can be used to convert logarithmic equations to exponential form.
• Exponential and logarithmic equations can be solved using algebraic techniques.

Final Answer

The final answer is $\boxed{C}$, which corresponds to the answer choice ${5^5=x}$.

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

A: An exponential equation is an equation that involves an exponential expression, such as $a^x = b$, where $a$ is the base and $x$ is the exponent. A logarithmic equation, on the other hand, is an equation that involves a logarithmic expression, such as $\log_a b = c$, where $a$ is the base and $c$ is the logarithm.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms, such as the product rule, the quotient rule, and the power rule. For example, if you have the equation $\log_a (bc) = c$, you can use the product rule to rewrite it as $\log_a b + \log_a c = c$.

Q: How do I convert a logarithmic equation to exponential form?

A: To convert a logarithmic equation to exponential form, you can use the definition of logarithms, which states that if $\log_a b = c$, then $a^c = b$. For example, if you have the equation $\log_5 x = 3$, you can rewrite it as $5^3 = x$.

Q: What is the difference between a base and an exponent?

A: The base is the number that is being raised to a power, while the exponent is the power to which the base is being raised. For example, in the equation $2^3 = 8$, the base is 2 and the exponent is 3.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use algebraic techniques, such as isolating the variable and using inverse operations. For example, if you have the equation $2^x = 8$, you can rewrite it as $2^x = 2^3$, and then use the fact that if $a^x = a^y$, then $x = y$.

Q: What is the relationship between exponential and logarithmic equations?

A: Exponential and logarithmic equations are inverse operations, meaning that they "undo" each other. For example, if you have the equation $2^x = 8$, you can rewrite it as $\log_2 8 = x$, which is a logarithmic equation.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you can use a calculator or a table of exponential values. For example, if you have the expression $2^3$, you can use a calculator to find that it is equal to 8.

Q: What are some common exponential and logarithmic equations?

A: Some common exponential and logarithmic equations include:

• $a^x = b$
• $\log_a b = c$
• $2^x = 8$
• $\log_2 8 = x$
• $5^x = 25$
• $\log_5 25 = x$

Q: How do I use exponential and logarithmic equations in real-life situations?

A: Exponential and logarithmic equations are used in a wide range of real-life situations, including:

• Finance: Exponential and logarithmic equations are used to calculate interest rates and investment returns.
• Science: Exponential and logarithmic equations are used to model population growth and decay, as well as to describe the behavior of physical systems.
• Engineering: Exponential and logarithmic equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Q: What are some common mistakes to avoid when working with exponential and logarithmic equations?

A: Some common mistakes to avoid when working with exponential and logarithmic equations include:

• Confusing the base and the exponent
• Failing to simplify logarithmic expressions
• Using the wrong property of logarithms
• Failing to check the domain of the equation

Q: How do I practice solving exponential and logarithmic equations?

A: To practice solving exponential and logarithmic equations, you can try the following:

• Work through practice problems in a textbook or online resource
• Use a calculator or a table of exponential values to evaluate expressions
• Try solving equations with different bases and exponents
• Use real-life examples to illustrate the use of exponential and logarithmic equations.