Solve $4^{x-3}=18$. Round To The Nearest Thousandth.

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Introduction

In this article, we will be solving an exponential equation of the form $4^{x-3}=18$. This type of equation involves a variable in the exponent, and our goal is to isolate the variable and find its value. We will use various mathematical techniques and properties of exponents to solve this equation.

Understanding Exponential Equations

Exponential equations are equations that involve a variable in the exponent. They have the form $a^{x}=b$, where $a$ is a base and $b$ is an exponent. In our equation $4^{x-3}=18$, the base is $4$ and the exponent is $x-3$. Our goal is to find the value of $x$ that satisfies this equation.

Isolating the Variable

To solve this equation, we need to isolate the variable $x$. We can do this by using the properties of exponents. Specifically, we can use the property that states $a^{x}=b \implies x = \log_{a}b$, where $\log_{a}b$ is the logarithm of $b$ to the base $a$. We can apply this property to our equation by taking the logarithm of both sides.

Taking the Logarithm of Both Sides

Taking the logarithm of both sides of the equation $4^{x-3}=18$ gives us:

log⁡4(4x−3)=log⁡4(18)\log_{4}(4^{x-3}) = \log_{4}(18)

Using the property of logarithms that states $\log_{a}(a^{x}) = x$, we can simplify the left-hand side of the equation:

x−3=log⁡4(18)x-3 = \log_{4}(18)

Solving for x

Now we can solve for $x$ by adding $3$ to both sides of the equation:

x=log⁡4(18)+3x = \log_{4}(18) + 3

Evaluating the Logarithm

To evaluate the logarithm $\log_{4}(18)$, we can use the change of base formula, which states that $\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$ for any positive real numbers $a$, $b$, and $c$, where $c \neq 1$. We can choose $c = 10$ as our base, since it is a common base for logarithms.

log⁡4(18)=log⁡10(18)log⁡10(4)\log_{4}(18) = \frac{\log_{10}(18)}{\log_{10}(4)}

Calculating the Logarithm Values

Using a calculator, we can find the values of the logarithms:

log⁡10(18)≈1.2553\log_{10}(18) \approx 1.2553

log⁡10(4)≈0.6021\log_{10}(4) \approx 0.6021

Substituting the Logarithm Values

Substituting these values into our equation for $x$, we get:

x≈1.25530.6021+3x \approx \frac{1.2553}{0.6021} + 3

Evaluating the Expression

Evaluating this expression, we get:

x≈2.0833+3x \approx 2.0833 + 3

x≈5.0833x \approx 5.0833

Rounding to the Nearest Thousandth

Rounding this value to the nearest thousandth, we get:

x≈5.083x \approx 5.083

Conclusion

In this article, we solved the exponential equation $4^{x-3}=18$ by isolating the variable $x$ and using the properties of logarithms. We evaluated the logarithm using the change of base formula and calculated the value of $x$ to the nearest thousandth.

Final Answer

The final answer is: 5.083\boxed{5.083}

Introduction

In our previous article, we solved the exponential equation $4^{x-3}=18$ by isolating the variable $x$ and using the properties of logarithms. In this article, we will answer some common questions related to solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It has the form $a^{x}=b$, where $a$ is a base and $b$ is an exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$. You can do this by using the properties of exponents, such as the property that states $a^{x}=b \implies x = \log_{a}b$, where $\log_{a}b$ is the logarithm of $b$ to the base $a$.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to change the base of a logarithm. It states that $\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}$ for any positive real numbers $a$, $b$, and $c$, where $c \neq 1$.

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you can use a calculator or a logarithm table. Alternatively, you can use the change of base formula to change the base of the logarithm to a more familiar base, such as base 10.

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

A: A logarithmic equation is an equation that involves a variable as the exponent of a logarithm, while an exponential equation is an equation that involves a variable as the exponent of a number. For example, $\log_{a}(x)=b$ is a logarithmic equation, while $a^{x}=b$ is an exponential equation.

Q: Can I solve an exponential equation using algebraic methods?

A: In some cases, you can solve an exponential equation using algebraic methods, such as factoring or quadratic formula. However, in general, exponential equations require the use of logarithms to solve.

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

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

  • Forgetting to isolate the variable $x$
  • Using the wrong base for the logarithm
  • Not evaluating the logarithm correctly
  • Not rounding the answer to the correct number of decimal places

Q: How do I round my answer to the correct number of decimal places?

A: To round your answer to the correct number of decimal places, you need to evaluate the logarithm to the correct number of decimal places and then round the result.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. In fact, calculators are often the most convenient way to solve exponential equations, especially when the base is a large number.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling electrical circuits

Conclusion

In this article, we answered some common questions related to solving exponential equations. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in solving exponential equations.

Final Answer

The final answer is: 5.083\boxed{5.083}