Problem 48 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log x=3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1000\). No decimal approximation is required as the answer is already a whole number.

Step 1: Identify the base of logarithm

Since the base of the logarithm is not specified, we assume it to be base 10, as per the convention for logarithms.

Step 2: Convert the logarithmic equation to exponential form

The logarithm equation can be converted to an exponential form using the rule \(b^{a}=x\), where \(b\) is the base of the logarithm, \(a\) is the logarithm, and \(x\) is the number. In this case, our equation \(\log x=3\) becomes \(10^{3}=x\).

Step 3: Solve for \(x\)

When we raise 10 to the power of 3, we get \(x = 1000\).

Step 4: Check if \(x\) is in the domain of original logarithmic expression

The logarithmic function is not defined for 0 and negative numbers. In this case, \(x = 1000\) which is a positive number and thus it lies in the domain of the original logarithmic expression and is a valid solution.

Step 5: Provide approximate decimal value

The exact value of \(x\) is already a whole number, \(x=1000\), so no further approximation is required.

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is crucial for solving logarithmic equations effectively. Logarithms, abbreviated as 'log', have several rules that simplify complex algebraic expressions. One primary property is the logarithm's base which, when not specified, is commonly assumed to be 10 (called the common logarithm).

Another key logarithm property involves the relationship between logarithms and exponents. This is showcased in the identity \( log_b(a) = c \) if and only if \( b^c = a \). This relationship is essential in converting logarithmic forms to their corresponding exponential forms, as its application often simplifies the steps needed to solve for variables.

Additional properties include the product rule \( log_b(xy) = log_b(x) + log_b(y) \), the quotient rule \( log_b(x/y) = log_b(x) - log_b(y) \) and the power rule \( log_b(x^c) = c\cdot log_b(x) \). Understanding these properties allows students to break down complex equations into more manageable parts, facilitating easier problem-solving.

Exponential Form Conversion

In solving logarithmic equations, it's necessary to be comfortable with converting between log form and exponential form. This skill is particularly valuable for isolating the exponent in an equation, as demonstrated in the example equation \(\log x = 3\). When we factor in the implied base of 10, we convert this log equation to its exponential counterpart \(10^3 = x\).

Here's how the conversion process generally works: you take the base of the logarithm (often 10 if unspecified) and raise it to the power of the number on the other side of the equation. The result is then set equal to the number that was inside the log. This process is derived from the fundamental definition of a logarithm, which serves as a way to find the exponent that a certain base must be raised to in order to obtain a given number.

Exponential form representation is more intuitive for many because it links directly with the basic arithmetic operation of exponentiation that they are generally comfortable with. Being adept at converting to exponential form can thus significantly streamline the process of solving logarithmic equations.

Logarithm Domain

A critical aspect of working with logarithms is understanding their domain. The domain of a logarithmic function contains all the input values (typically denoted as \( x \)) for which the function is defined. In general, logarithms have a specific limitation: they are only defined for positive numbers. In mathematics, this principle is because taking the log of non-positive numbers does not produce a real number result.

This restriction affects how we approach solving log equations—any solution we find must be within the domain of the original logarithmic expressions. For instance, solving \(\log x = 3\) yields \(x = 1000\), as per the steps provided. It is imperative to then verify that \(x\) falls within the domain. Since \(1000\) is positive, it satisfies the condition, confirming its validity as a solution.

Always remember, when solving for \(x\) in a logarithmic equation, the solution must be positive. Any negative or zero value does not lie within the domain of a logarithm and must be rejected, ensuring the mathematical integrity of the solution.