Problem 11 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=2.6143 $$

Step-by-Step Solution

Answer

\( x \approx 411.15 \)

Step 1: Understand the Problem

We are given the logarithm of a number and need to find the number itself. Specifically, we are told that \( \log x = 2.6143 \). Our task is to determine the value of \( x \) using a calculator.

Step 2: Recall the Definition of Logarithm

The expression \( \log x = 2.6143 \) means that \( x \) is equal to \( 10^{2.6143} \). This is based on the definition of a common logarithm, which is log base 10.

Step 3: Evaluate Using a Calculator

Use a calculator to find \( 10^{2.6143} \). Enter the exponent into the calculator and compute the result to find \( x \).

Step 4: Round to Five Significant Digits

Once you have computed the value from the calculator, round the answer to five significant digits for precision. This step ensures the answer meets the problem's requirements.

Key Concepts

Common Logarithms

Logarithms are a fascinating mathematical concept that serve as the inverse operation to exponentiation. Although there are various types of logarithms, common logarithms use the base 10, making them a cornerstone in computations pertaining to exponential growth or decay. This particular kind of logarithm is often simply written as \( \log x \), where the base 10 is implied.

To clarify, the statement \( \log x = 2.6143 \) means that \( x \) is the number which 10 must be raised to in order to get \( x \). In formulaic terms, this translates to the power equation \( x = 10^{2.6143} \). This understanding links logarithms directly to exponents, a crucial connection for algebra and higher mathematics. Whenever you see "log" without a base, you can confidently assume it's a common logarithm.

Using a calculator allows you to easily perform this calculation, providing the precise value for \( x \), which in this problem, after calculation and rounding, achieves the required significant figures.

Exponential Functions

Exponential functions are vital in representing relationships in which growth or decay happens at a constantly proportional rate to the current value. The general form of an exponential equation is \( y = a \cdot b^x \), where \( a \) is a constant and \( b \) is the base of the exponential.

In the context of our original problem, understanding that \( x = 10^{2.6143} \) underlines the application of exponential functions. Finding the value of \( x \) is the process of evaluating the exponential function with base 10, which is crucial for comprehending anything from scientific notation to financial interest calculations.

Key points about exponential functions:

• They increase (or decrease) rapidly and consistently.
• The base (here 10) determines the rate of growth or decay.
• They are often used to model real-world phenomena, including population growth, radioactive decay, and interest compounding.

Understanding how to work with these functions is fundamental to solving complex real-world problems.

Significant Figures

Significant figures are a way to express precision in numerical results, crucial for maintaining accuracy within a scientific context. They indicate which digits in a number are meaningful, contributing to its precision.

In the context of solving logarithmic equations, rounding the result to a specified number of significant digits, like in the exercise where we round to five significant digits, ensures a balance between correctness and simplicity. This practice is essential not just in plain computation but in expressing real-world data, where measurements have a degree of uncertainty.

Here are some basic rules for significant figures:

• All non-zero digits are always significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.

Understanding and utilizing significant figures are especially important in chemistry, physics, and any field that relies on quantitative analysis to ensure the results have the desired precision.