Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$10^{x}=3.91$$

Step-by-Step Solution

Answer

The solution to the equation \(10^{x} = 3.91\) is approximately \(x = 0.59\), correct to two decimal places.

Step 1: Apply Logarithm to Both Sides

Apply the logarithm to both sides of the equation to simplify the equation.\n Applying the log function to both sides of the equation \(10^{x} = 3.91\), we get \(\log(10^{x}) = \log(3.91)\).

Step 2: Use Basic Logarithm Law

Use the logarithmic law to simplify further. Using the law \(\log(a^{b}) = b * \log(a)\), the equation \(\log(10^{x}) = \log(3.91)\) simplifies to \(x * \log(10) = \log(3.91)\). Since \(\log(10) = 1\), the equation further simplifies to \(x = \log(3.91)\).

Step 3: Compute Decimal Approximation

Using a calculator, calculate the value of \(\log(3.91)\) to get the decimal approximation of \(x\). The decimal approximation of \(x\) is approximately 0.59 when rounded to two decimal places.

Key Concepts

Logarithmic Functions

In the realm of mathematics, logarithmic functions are the inverse operations of exponentiation. This means that they are used to determine the power to which a number, called the base, must be raised to produce a given number. For example, the logarithm base 10 of 1000 is 3, because 10 raised to the power of 3 equals 1000. In general form, this is written as \( \log_{b}(x) = y \) if and only if \( b^y = x \).
In our exercise, with the equation \(10^{x} = 3.91\), taking the logarithm of both sides essentially means finding the power \(x\) to which 10 must be raised to result in 3.91. The base of the logarithm corresponds to the base of the exponent. Here, we are working with base 10, often referred to as the common logarithm and denoted as \( \log(x) \).

Why Use Logarithms?

Logarithms simplify the process of solving exponential equations by converting multiplication into addition, division into subtraction, and exponentiation into multiplication. This transformation is essential in solving equations where the variable is an exponent, as it brings the variable down to a level where we can solve for it linearly.

Decimal Approximation

Decimal approximation is a powerful mathematical tool used for representing a number in a more simplified and usable form. It often involves rounding numbers to a specific number of decimal places for ease of computation or interpretation. In practice, when calculators or computers provide us with a long string of digits, we frequently approximate these to a certain number of decimal places to make the numbers more manageable.
In the context of our exercise, once the equation has been simplified to \( x = \log(3.91) \), we seek a decimal approximation of the solution. Calculators generally use the common logarithm base 10, which means when we enter \( \log(3.91) \), we'll get a decimal result. If we're advised to round our answer to two decimal places, the calculator's display of the result as '0.59176...' would be approximated and written as \( x \approx 0.59 \).

Real-World Importance

Decimal approximation isn't just a mathematical exercise; it's highly relevant in everyday life. In fields such as engineering, economics, and science, obtaining precise decimal approximations quickly can be crucial for making decisions or measurements that are practical and applicable.

Properties of Logarithms

Logarithms have unique properties that allow us to manipulate and solve logarithmic equations efficiently. One such property is \( \log(a^{b}) = b * \log(a) \), which we utilize in step 2 of our exercise to simplify \( \log(10^{x}) = \log(3.91) \).
This property tells us that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base number. Essentially, it helps us bring down the exponent, making the variable easier to isolate and solve for, which is particularly handy in our original equation with the variable in the exponent.

Other Noteworthy Properties

• Product rule: \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \)
• Quotient rule: \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \)
• Change of base formula: \( \log_{b}(x) = \frac{\log_{c}(x)}{\log_{c}(b)} \) for any positive base \(c\), different from 1

These properties make the logarithm a flexible and indispensable tool in algebra, calculus, and beyond, especially when dealing with complex exponential equations.