Problem 15 - Algebra for College Students

Question

Use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=1.9006 $$

Step-by-Step Solution

Answer

x is approximately 79.433.

Step 1: Understand the problem

We are given the logarithm of a number, \(\log x = 1.9006\), and need to find the actual number \(x\). Since no base is specified, we assume this is a common logarithm, which has a base of 10.

Step 2: Apply the logarithmic property

To find \(x\), we need to transform the logarithmic equation into an exponential equation. The property \(\log_{10} x = y\) implies that \(x = 10^y\). Thus, we have the result as \(x = 10^{1.9006}\).

Step 3: Calculate the exponential value using a calculator

Enter the expression \(10^{1.9006}\) into a calculator to solve for \(x\). This will give you the numerical value of \(x\).

Step 4: Round the result to five significant digits

After calculating \(10^{1.9006}\) on a calculator, make sure to round the result to five significant digits. The calculator gives \(x \approx 79.4328\), and when rounded to five significant digits, it is \(79.433\).

Key Concepts

Common Logarithms

Common logarithms are logarithms with a base of 10. When we see \(\log x\) without a specified base, it's automatically assumed to be a common logarithm. This is why in the step by step solution for \(\log x = 1.9006\), the base was considered 10. Common logarithms are widely used in scientific and engineering applications for their simplicity, as they tie directly into our decimal-based number system.
Benefits of common logarithms include:

• Ease of use with simple calculators.
• Direct relevance to measurements and scientific scales, such as the Richter scale for earthquakes.

Understanding the base of a logarithm is crucial to correctly interpreting and solving log equations.

Exponential Equations

Exponential equations involve variables in the exponent and are solved by using logarithms to bring down the power. In our exercise, we used the property of logarithms to turn the given form \(\log_{10} x = 1.9006\) into an equivalent exponential form \(x = 10^{1.9006}\).
Understanding these transformations is essential because:

• It allows us to move between logarithmic and exponential forms.
• It helps in solving equations where the unknown is an exponent.

This conversion is a fundamental process in algebra, allowing complex problems to become solvable using basic arithmetic operations.

Significant Digits

Significant digits in a number represent the precision of the measurement or calculation. In our exercise, rounding the result of the exponential expression \(10^{1.9006}\) to five significant digits ensures the accuracy and reliability of our result. The calculated value of approximately 79.4328 was rounded to 79.433 to maintain precision.
Important aspects include:

• Ensuring consistency of precision across calculations.
• Avoiding the exaggeration of precision in scientific reporting.

Recognizing and using significant digits is critical for maintaining clarity and accuracy in scientific communication.

Calculator Usage

Calculators are valuable tools for solving equations involving logarithms and exponents. They perform complex calculations quickly and precisely, which is crucial for achieving accurate results like in our example, where \(x = 10^{1.9006}\) was calculated as 79.4328.
Tips for effective calculator use:

• Familiarize yourself with calculator functions, such as the \(10^x\) button for calculating powers of ten.
• Double-check each input to reduce errors.

Using calculators wisely can save time and improve accuracy in mathematical tasks.