Problem 18 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-0.1452 $$

Step-by-Step Solution

Answer

\(x \approx 0.71346\).

Step 1: Understanding the Given Logarithm Equation

In this exercise, you're given the logarithmic equation \(\log x = -0.1452\). This indicates the logarithm with base 10. Your task is to find the value of \(x\) that satisfies this equation.

Step 2: Converting Logarithmic Equation to Exponential Form

To solve for \(x\), convert the logarithmic equation into its exponential form using the relationship \(\log_{10} x = y \Rightarrow 10^y = x\). Therefore, the exponential form is \(x = 10^{-0.1452}\).

Step 3: Calculating the Exponential Value

Use a calculator to compute \(10^{-0.1452}\). Enter the base (10), exponent (-0.1452), and use the power function of your calculator to find the exact value of \(x\).

Step 4: Expressing the Solution to Significant Figures

The problem asks for the answer to be expressed to five significant digits. After calculating \(10^{-0.1452}\), round the value to five significant digits.

Key Concepts

Exponential Equations

In mathematics, exponential equations are equations in which variables appear as exponents. Solving these equations often involves logarithms, which help to "lower" the exponents for easier manipulation.
An example is the equation \(10^y = x\). Here, \(x\) is an exponential equation because the variable \(x\) is connected with the base ten through an exponent \(y\). To find \(x\), we can use the inverse operation of logarithms. Given the problem \(\log x = -0.1452\), we rewrite it as \(10^{-0.1452} = x\).
Understanding this relationship between logarithms and exponents is crucial for solving various mathematical problems efficiently, making exponential equations a fundamental component of advanced math topics.

Calculator Usage

Solving exponential equations often requires the use of a scientific calculator. This can seem daunting at first, but with a few tips, you can master these calculations easily.
Firstly, understand the key functions of your calculator:

• **Power Function**: Used to calculate powers, often represented as \("^\"\) or as a specific button like "EXP" or "POWER".
• **Logarithm Function**: Usually appears as "LOG" for base 10 logs.

For solving equations like \(10^{-0.1452} = x\), follow these steps:
1. **Enter the base** (10).2. **Use the exponentiation function** to input the power, -0.1452.
Ensure you're using the correct function to get an accurate result, which will help you find \(x\) quickly and precisely.

Significant Figures

Expressing answers in significant figures is essential for precision in scientific and mathematical calculations. Significant figures reflect the accuracy of a measurement or calculation.
For the given problem, the answer must be expressed to five significant digits, ensuring the solution maintains its precision. When you compute \(10^{-0.1452}\), the calculator will give you a number with many decimal places.

• **Rounding Rules**: Decide based on the next digit; if it's 5 or more, round up.Otherwise, round down.
• **Counting Significant Figures**: Start from the first non-zero digit to the last digit being considered significant.

Using this approach guarantees that your final solution is both precise and reliable. For instance, if the solution is 0.713838...To five significant figures, it would be 0.71384.