Solve For { X $}$ In The Equation { 4^{x-3} = 18 $} . R O U N D Y O U R A N S W E R T O T H E N E A R E S T T H O U S A N D T H . . Round Your Answer To The Nearest Thousandth. . R O U N D Yo U R An S W Er T O T H E N E A Res Tt H O U S An D T H . {$ X \approx \square $}$

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Introduction

In this article, we will delve into solving an exponential equation involving a base of 4. The equation given is { 4^{x-3} = 18 $}$, and we are tasked with finding the value of { x $}$ that satisfies this equation. We will use logarithmic properties to solve for { x $}$ and then round our answer to the nearest thousandth.

Understanding Exponential Equations

Exponential equations are equations that involve a base raised to a power. In this case, the base is 4, and the power is { x-3 $}$. The equation { 4^{x-3} = 18 $}$ can be read as "4 raised to the power of { x-3 $}$ equals 18".

Using Logarithms to Solve the Equation

To solve the equation { 4^{x-3} = 18 $}$, we can use logarithms. Specifically, we can use the logarithmic property that states { \log_b a^c = c \log_b a $}$. This property allows us to bring the exponent down and simplify the equation.

Applying the Logarithmic Property

Using the logarithmic property, we can rewrite the equation as:

{ \log_4 18 = x-3 $}$

Solving for { x $}$

Now that we have the equation in a simpler form, we can solve for { x $}$. To do this, we can add 3 to both sides of the equation:

{ \log_4 18 + 3 = x $}$

Evaluating the Logarithm

To evaluate the logarithm { \log_4 18 $}$, we can use a calculator or a logarithmic table. The value of { \log_4 18 $}$ is approximately 2.243.

Substituting the Value of the Logarithm

Now that we have the value of the logarithm, we can substitute it into the equation:

{ 2.243 + 3 = x $}$

Solving for { x $}$

Simplifying the equation, we get:

{ x = 5.243 $}$

Rounding the Answer

Finally, we are asked to round our answer to the nearest thousandth. Rounding 5.243 to the nearest thousandth gives us:

{ x \approx 5.243 $}$

Conclusion

In this article, we solved the equation { 4^{x-3} = 18 $}$ using logarithmic properties. We used the logarithmic property to bring the exponent down and simplify the equation, and then solved for { x $}$. Finally, we rounded our answer to the nearest thousandth. The value of { x $}$ that satisfies the equation is approximately 5.243.

Additional Tips and Tricks

  • When solving exponential equations, it's often helpful to use logarithms to bring the exponent down and simplify the equation.
  • Make sure to use the correct base when evaluating logarithms.
  • When rounding answers, make sure to round to the correct number of decimal places.

Frequently Asked Questions

  • Q: What is the value of { x $}$ that satisfies the equation { 4^x-3} = 18 $}$? A The value of { x $$ that satisfies the equation is approximately 5.243.
  • Q: How do I solve exponential equations? A: To solve exponential equations, you can use logarithmic properties to bring the exponent down and simplify the equation.
  • Q: What is the correct base to use when evaluating logarithms? A: The correct base to use when evaluating logarithms is the base of the logarithm, in this case, 4.

Introduction

In our previous article, we solved the equation { 4^{x-3} = 18 $}$ using logarithmic properties. In this article, we will answer some frequently asked questions about solving exponential equations.

Q&A

Q: What is the value of { x $}$ that satisfies the equation { 4^{x-3} = 18 $}$?

A: The value of { x $}$ that satisfies the equation is approximately 5.243.

Q: How do I solve exponential equations?

A: To solve exponential equations, you can use logarithmic properties to bring the exponent down and simplify the equation. Specifically, you can use the logarithmic property that states { \log_b a^c = c \log_b a $}$.

Q: What is the correct base to use when evaluating logarithms?

A: The correct base to use when evaluating logarithms is the base of the logarithm, in this case, 4.

Q: Can I use any base when solving exponential equations?

A: No, you should use the base that is the same as the base of the exponential expression. In this case, the base is 4.

Q: How do I round my answer to the nearest thousandth?

A: To round your answer to the nearest thousandth, you should look at the digit in the ten-thousandths place. If it is 5 or greater, you should round up. If it is less than 5, you should round down.

Q: What if I have a negative exponent?

A: If you have a negative exponent, you can rewrite the equation as { b^{-c} = \frac{1}{b^c} $}$. Then, you can use logarithmic properties to solve for the variable.

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

A: Yes, you can use a calculator to solve exponential equations. However, make sure to use the correct base and to round your answer to the correct number of decimal places.

Q: What if I have a fraction as an exponent?

A: If you have a fraction as an exponent, you can rewrite the equation as { b^{\frac{c}{d}} = (bc){\frac{1}{d}} $}$. Then, you can use logarithmic properties to solve for the variable.

Additional Tips and Tricks

  • When solving exponential equations, make sure to use the correct base and to round your answer to the correct number of decimal places.
  • If you have a negative exponent, you can rewrite the equation as { b^{-c} = \frac{1}{b^c} $}$.
  • If you have a fraction as an exponent, you can rewrite the equation as { b^{\frac{c}{d}} = (bc){\frac{1}{d}} $}$.

Conclusion

In this article, we answered some frequently asked questions about solving exponential equations. We covered topics such as the correct base to use when evaluating logarithms, how to round answers to the nearest thousandth, and how to handle negative and fractional exponents. We hope that this article has been helpful in answering your questions about solving exponential equations.

Further Reading

  • For more information on logarithmic properties, see our article on logarithmic properties.
  • For more information on solving exponential equations, see our article on solving exponential equations.
  • For more information on rounding numbers, see our article on rounding numbers.

Related Articles

  • Logarithmic Properties: A Guide to Solving Exponential Equations
  • Solving Exponential Equations: A Step-by-Step Guide
  • Rounding Numbers: A Guide to Rounding to the Nearest Thousandth