Problem 13 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=4.9547 $$

Step-by-Step Solution

Answer

\( x \approx 90205 \)

Step 1: Understand Logarithmic Equation

The given equation is \( \log x = 4.9547 \). This means that \( x \) is the number such that its base-10 logarithm is 4.9547. Essentially, we need to find \( x \) given this base-10 logarithm value.

Step 2: Apply Exponentiation Property

To solve for \( x \), we use the property of logarithms that states if \( \log_{10} x = a \), then \( x = 10^a \). Therefore, here we need to compute \( x = 10^{4.9547} \).

Step 3: Use a Calculator

On a scientific calculator, enter the expression \( 10^{4.9547} \). This operation calculates the antilogarithm or exponential value of \( 4.9547 \) in base 10.

Step 4: Get and Round the Result

After performing the calculation on the calculator, the result you get is approximately 90205.35489. Round this number to five significant digits to match the required precision.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number called the base to the power of an exponent. It is expressed as \(b^n\), where \(b\) is the base and \(n\) is the exponent. This operation is the opposite of taking a logarithm.
For example, when we compute \(10^{4.9547}\), we are performing exponentiation with 10 as the base, raised to the power of 4.9547. This transforms the problem of a given logarithm \(\log x = 4.9547\) into calculating \(x = 10^{4.9547}\).

• This process consists of repeatedly multiplying the base by itself according to the value of the exponent.
• Exponentiation is crucial for solving logarithmic equations, as it simplifies the finding of the original number \(x\) from its logarithm.

Using a calculator efficiently allows handling such calculations with ease, giving you precise results that are essential in real-world applications.

Antilogarithm

An antilogarithm is the inverse function of a logarithm. While a logarithm gives you the exponent or power that a base number is raised to reach a certain value, the antilogarithm allows you to find this original numerical value given the exponent.
In our problem, we find \(x\) by taking the antilogarithm of 4.9547, essentially computing the exponential value of this logarithm by determining \(10^{4.9547}\).
This is why we use a calculator to find the antilogarithm, ensuring that the operation is accurate and efficient:

• Calculate the antilogarithm by entering the value as the exponent of the base on your calculator.
• This operation gives you the numerical value of \(x\), resulting in the original number that had the logarithm of 4.9547.

Antilogarithms are particularly useful in situations where quick and accurate recalculations of quantifiable data are necessary.

Significant Digits

Significant digits are the numbers in a value that contribute to its precision. They exclude any leading or trailing zeroes unless these zeroes are sandwiched between other significant figures.
For our exercise, where we computed \(x = 10^{4.9547}\), the result of approximately 90205.35489 was rounded to five significant digits, yielding 90205. This reflects the accuracy of the calculation based on significant figures and helps communicate precision clearly.
Guidelines for determining significant digits include:

• Start counting from the first non-zero digit.
• All numbers after this starting point are considered significant until the end of the decimal.

Rounding to the specified number of significant digits is essential in science and engineering to ensure that results are both accurate and usable in further calculations.