Solve For $x$. Give An Exact Answer In Base 10 And An Approximate Answer To The Nearest Hundredth (0.01).$5^{x-3} - 3 = 21$

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Solve for x: A Step-by-Step Guide to Finding the Exact and Approximate Values of x

In this article, we will delve into the world of mathematics and solve for x in the given equation: $5^{x-3} - 3 = 21$. We will start by understanding the equation, then proceed to solve for x using algebraic manipulations and logarithmic properties. Finally, we will provide the exact answer in base 10 and an approximate answer to the nearest hundredth (0.01).

The given equation is $5^{x-3} - 3 = 21$. To solve for x, we need to isolate the variable x on one side of the equation. The first step is to add 3 to both sides of the equation to get rid of the negative term.

5x−3=245^{x-3} = 24

Now that we have isolated the exponential term, we can take the logarithm of both sides to get rid of the exponent. We will use the logarithmic property that states: $\log_a{b^c} = c \cdot \log_a{b}$

Taking the logarithm of both sides, we get:

log⁡55x−3=log⁡524\log_5{5^{x-3}} = \log_5{24}

Using the logarithmic property, we can simplify the left-hand side of the equation:

(x−3)⋅log⁡55=log⁡524(x-3) \cdot \log_5{5} = \log_5{24}

Since $\log_a{a} = 1$, we can simplify the equation further:

(x−3)⋅1=log⁡524(x-3) \cdot 1 = \log_5{24}

Now that we have simplified the equation, we can solve for x. We will start by isolating the variable x on one side of the equation.

x−3=log⁡524x-3 = \log_5{24}

Adding 3 to both sides of the equation, we get:

x=log⁡524+3x = \log_5{24} + 3

To find the exact value of x, we need to evaluate the logarithmic expression: $\log_5{24}$

Using a calculator or a logarithmic table, we can find that:

log⁡524≈2.585\log_5{24} \approx 2.585

Substituting this value into the equation, we get:

x≈2.585+3x \approx 2.585 + 3

x≈5.585x \approx 5.585

To find the approximate value of x, we can round the exact value to the nearest hundredth (0.01).

x≈5.59x \approx 5.59

In this article, we solved for x in the given equation: $5^{x-3} - 3 = 21$. We started by understanding the equation, then proceeded to solve for x using algebraic manipulations and logarithmic properties. Finally, we provided the exact answer in base 10 and an approximate answer to the nearest hundredth (0.01).

Exact Value of x

x=log⁡524+3x = \log_5{24} + 3

Approximate Value of x

x≈5.59x \approx 5.59

Final Answer

The final answer is: 5.59\boxed{5.59}
Solve for x: A Q&A Guide to Understanding the Solution

In our previous article, we solved for x in the given equation: $5^{x-3} - 3 = 21$. We provided the exact answer in base 10 and an approximate answer to the nearest hundredth (0.01). In this article, we will answer some frequently asked questions about the solution to help you better understand the concept.

Q: What is the exact value of x?

A: The exact value of x is $\log_5{24} + 3$.

Q: How do I evaluate the logarithmic expression?

A: You can use a calculator or a logarithmic table to evaluate the logarithmic expression. Alternatively, you can use the change of base formula to convert the logarithm to a common base.

Q: What is the approximate value of x?

A: The approximate value of x is 5.59.

Q: Why do we need to add 3 to the logarithmic expression?

A: We need to add 3 to the logarithmic expression because of the initial equation $5^{x-3} - 3 = 21$. When we add 3 to both sides of the equation, we get $5^{x-3} = 24$. This allows us to isolate the exponential term and take the logarithm of both sides.

Q: Can I use a different base for the logarithm?

A: Yes, you can use a different base for the logarithm. However, you need to be careful when changing the base, as it can affect the value of the logarithm.

Q: How do I round the exact value of x to the nearest hundredth?

A: To round the exact value of x to the nearest hundredth, you need to look at the third decimal place. If the third decimal place is 5 or greater, you round up. If the third decimal place is less than 5, you round down.

Q: What is the significance of the logarithmic expression?

A: The logarithmic expression $\log_5{24}$ represents the power to which the base 5 must be raised to obtain the number 24. In this case, the logarithmic expression is approximately 2.585.

In this article, we answered some frequently asked questions about the solution to the equation $5^{x-3} - 3 = 21$. We provided the exact and approximate values of x, as well as explanations for the steps involved in solving the equation. We hope this Q&A guide has helped you better understand the concept and has provided you with a deeper understanding of logarithmic expressions.

Final Answer

The final answer is: 5.59\boxed{5.59}