Problem 97 - Elementary and Intermediate Algebra

Question

Use a calculator to solve each equation. Round answers to four decimal places. See Example \(6 .\) $$ \log x=-1.71 $$

Step-by-Step Solution

Answer

\( x \approx 0.0195 \)

Step 1: Understand the Logarithmic Equation

The given equation \( \log x = -1.71 \) means we are using the common logarithm (base 10). We want to find the value of \( x \) such that its logarithmic value equals \(-1.71\).

Step 2: Use the Definition of Logarithms

The equation \( \log x = -1.71 \) means \( x = 10^{-1.71} \). This is because the logarithmic function \( \log x \), with base 10, is the inverse of the exponential function \( 10^y \).

Step 3: Calculate the Exponential Value

Use a calculator to compute \( 10^{-1.71} \). Enter \(-1.71\) and then press the \( 10^x \) or equivalent exponential function button on your calculator. The result is approximately \( 0.0195 \).

Step 4: Round the Answer

The calculated value of \( 10^{-1.71} \) is \( 0.0195455 \ldots \). Rounding this value to four decimal places, we get \( x \approx 0.0195 \).

Key Concepts

Understanding Common Logarithm

The common logarithm is a logarithm with base 10. When you see "log" without a base specified, it's usually the common logarithm. This is different from natural logarithms, which have a base of \( e \), approximated as 2.718.Common logarithms help us solve for exponents in equations where the base is 10. In the example equation, \( \log x = -1.71 \), the common logarithm tells us the power we need to raise 10 to get \( x \). If \( \log x = y \), then \( x \) equals \( 10^y \). Understanding this relationship is key to solving logarithmic equations.Using common logarithms, you can quickly find out how many times you need to multiply a base (10) to reach a certain number. This makes it a handy tool in calculations, especially when working with very large or small numbers.

The Power of Exponentiation

Exponentiation is the mathematical operation that involves raising a number to a power. It's the inverse operation of taking a logarithm. When solving \( \log x = -1.71 \), what we're eventually doing is exponentiation: converting \( -1.71 \) back into the original number \( x \) by calculating \( 10^{-1.71} \).Here are the steps once more for clarity:

• Identify the equation: \( \log x = -1.71 \).
• Rewrite it by thinking about what amount of 10 you need to multiply to get \( x \), which is done by exponentiation: \( x = 10^{-1.71} \).
• Use a calculator to compute this power, finding \( x \).

Exponentiation is essential, especially when dealing with equations involving growth or decay, such as calculating compound interest or exponential decay in physics.When converting back and forth between logarithms and exponentials, calculators are invaluable for computing these conversions quickly.

Rounding Decimal Places for Precision

Rounding is a mathematical process to simplify numbers. It helps present data in a more manageable form without significant loss of accuracy. When dealing with logarithmic equations, precision is paramount, and sometimes you will need to limit the number of decimal places.In the solution of \( 10^{-1.71} \), the calculated value was approximately 0.0195455. However, we require the answer rounded to four decimal places, giving us 0.0195. Here is a step-by-step guide to rounding to four decimal places:

• Identify the number in the fifth decimal place.
• If the fifth number is 5 or greater, increase the fourth decimal place by one.
• If the fifth number is less than 5, keep the fourth decimal as is.

Rounding makes it easier to work with numbers in calculations, especially when precision to a particular decimal place is required. It’s used in various fields, from engineering to finance, to ensure clarity and accuracy.