Problem 4 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. Check your answers. $$ 3^{x}=27.3 $$

Step-by-Step Solution

Answer

The value of \(x\) rounded off to the nearest ten-thousandth is approximately 3.0020, which is verified by substituting it back into the original equation. Please note that calculators may vary slightly due to rounding.

Step 1: Write the Equation in Logarithmic Form

Express the exponential equation \(3^{x} = 27.3 \) in its corresponding logarithmic form for easier comparison and calculation. In logarithmic form, the equation becomes \( \log_3 {27.3} = x \) or \( x = \log_3 {27.3} \).

Step 2: Calculate the Value of Logarithm

Utilize a calculator to find the logarithm base 3 of '27.3'. This will yield a decimal that might be precise up to many decimal places. Given the instruction in the problem, the answer has to be rounded off to the nearest ten-thousandth.

Step 3: Round off the Answer

After getting the decimal result, round off the value to the nearest ten-thousandth.

Step 4: Check the Answer

Substitute the rounded value of 'x' into the original equation and check to confirm it's accurate. The result should equal '27.3' when you substitute your answer back into the original equation \(3^{x} = 27.3 \).

Key Concepts

Logarithmic Form

When you encounter an equation like \(3^x = 27.3\), transforming it into logarithmic form makes solving for \(x\) much easier. Logarithms essentially "unwrap" the exponent. The logarithmic identity used here is:

• If \(b^y = a\), then \(y = \log_b a\).

For our equation, this becomes \(x = \log_3 27.3\). By expressing the equation this way, you shift the problem from direct calculation to one that is more manageable with logarithmic functions. This format utilizes the fact that logarithms are the inverse operations of exponentiation.

Using a calculator, you would solve for \(x\) with the log base 3 function, simplifying the process to handle calculations that would be more cumbersome in exponential form.

Rounding Decimals

After finding the logarithmic value \(\log_3 27.3\), you will get a long decimal number. In mathematics, the accuracy of your answer may need to be presented to a specific number of decimal places for practical reasons or as instructed.

• To round to the nearest ten-thousandth, look at the fifth decimal place.
• If this number is 5 or greater, increase the fourth decimal place by 1. If it's less than 5, keep the fourth decimal place as it is.

For example, if your result is 3.049872, you round it to 3.0499. Rounding ensures that your answers are easier to interpret and use in practical scenarios without significant loss of accuracy. It is a minor adjustment but can change results significantly if not done properly.

Checking Solutions

Checking your solution ensures that the answer you calculated for \(x\) is accurate. This is an essential step in solving mathematical equations, verifying that each calculation holds true to the original equation.

• Substitute your rounded decimal \(x\) back into the exponential equation \(3^x = 27.3\).
• Use a calculator to compute the left side using this value of \(x\).
• Check if it equals the original number, 27.3. Ideally, they should match very closely once rounded.

Through this process, you confirm the accuracy of your calculations and interpretations. If there is a large discrepancy, revisit your logarithm calculation and rounding steps to find potential mistakes.