Problem 78 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. (a) \(\log x=1\) (b) \(\log x=-4\) (c) \(\log x=0.3\)

Step-by-Step Solution

Answer

(a) 10, (b) 0.0001, (c) 2.0000

Step 1: Understand the Problem

We need to solve logarithmic equations, which means finding the value of \( x \) such that the equation is satisfied for each case. The logarithm with no specified base is assumed to be base 10.

Step 2: Solve \( \log x = 1 \)

To solve \( \log x = 1 \), rewrite it in exponential form: \( x = 10^1 \). Calculate the value: \( x = 10 \).

Step 3: Solve \( \log x = -4 \)

To solve \( \log x = -4 \), rewrite it in exponential form: \( x = 10^{-4} \). Calculate the value: \( x = 0.0001 \).

Step 4: Solve \( \log x = 0.3 \)

To solve \( \log x = 0.3 \), rewrite it in exponential form: \( x = 10^{0.3} \). Use a calculator to approximate the value: \( x \approx 2.0000 \) when rounded to four decimal places.

Key Concepts

Base 10 Logarithm

The concept of a logarithm can seem confusing at first, but it's all about understanding powers of numbers. A base 10 logarithm, indicated as \( \log x \) or \( \log_{10} x \), is simply the power that 10 must be raised to get the number \( x \). Think of it like asking, "How many times do I multiply 10 to get this number?"
To clarify:

• If \( \log x = 1 \), it means 10 must be raised to the power of 1 to equal \( x \). Hence, \( x = 10 \).
• If \( \log x = -4 \), then 10 must be raised to the power of -4 to equal \( x \), so \( x = 0.0001 \).
• If \( \log x = 0.3 \), then 10 raised to 0.3 gives \( x \).

Each time, you're finding what exponent on 10 gives you \( x \). This concept is used regularly in science and engineering to handle very large or small numbers.

Exponential Form

To solve logarithmic equations, it's helpful to convert them into exponential form. This is because generally, it's easier to understand numbers when they are in a simple, direct form.
With base 10 logarithms, the exponential form can be written as:

• \( \log x = a \) becomes \( x = 10^a \)

This transformation is crucial in solving logarithmic equations, since it allows you to rewrite the problem as a more straightforward mathematical operation.
For instance, the equation \( \log x = 0.3 \) converts to \( x = 10^{0.3} \). From here, we can use a calculator to determine that \( x \approx 2.0000 \) rounded to four decimal places. Understanding the switch between logarithmic and exponential forms helps when moving between different types of equations in mathematics.

Approximation

In mathematics, especially with non-integer logarithmic values, we often need approximations to reach a closer value. Calculators are particularly useful here.
When encountering terms like \( 10^{0.3} \), it is difficult to find an exact number by mental calculation. Instead, using a calculator will provide a numerical approximation:

• For \( \log x = 0.3 \), calculate \( 10^{0.3} \). The calculator gives approximately \( x = 1.9953 \), which we round to four decimal places \( x \approx 2.0000 \).

Approximations give hands-on solutions in real-life applications where exact numbers aren't always practical. They present a close enough value that's both achievable and useful. This process is essential in fields needing precision and accuracy like engineering, physics, and finance.