Problem 59 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{4}(3 x+2)=3 $$

Step-by-Step Solution

Answer

The exact solution to the equation is \(x = \frac{62}{3}\). The decimal approximation, correct to two decimal places, is \(x \approx 20.67\).

Step 1: Convert Logarithmic Equation to an Exponential Equation

Converting the logarithmic expression \(\log _{4}(3x+2)=3\) into its exponential form will give \(4^{3} = 3x + 2\). This conversion is achieved by remembering that for \(\log_b a = c\), the equivalent exponential form is \(b^{c} = a\).

Step 2: Simplify the Equation

Solving the equation \(4^{3} = 3x + 2\), we get \(64 = 3x + 2\). Next, isolate \(x\) by subtracting 2 from both sides of the equation to get \(62 = 3x\).

Step 3: Solve for x

Divide both sides of the equation \(62 = 3x\) by 3, we get \(x = \frac{62}{3}\).

Step 4: Check the Domain

The original logarithmic function is defined for values of \(x\) that make \(3x + 2 > 0\). Solving this for \(x\) yields \(x > -2/3\). Since the obtained solution, \(x = \frac{62}{3}\), is in this domain of the original expression, the solution is valid.

Step 5: Decimal Approximation

To get the decimal approximation of the exact answer, the quantity \(\frac{62}{3}\) can be calculated using a calculator to obtain \(x \approx 20.67\), correct to two decimal places.

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that include logarithms. A logarithm is the inverse operation of exponentiation and helps to find the power to which a base number is raised to obtain another number. For example, in the expression \( \log_4(3x+2) = 3 \), the base is 4, and the logarithm tells us the power (3 in this case) to which 4 must be raised to give the value \( 3x + 2 \).

Here are some important things to remember about logarithmic functions:

• The base of the logarithm must always be a positive number, except 1.
• Logarithmic expressions only take positive values of their argument (inside the log function).
• Logarithms are undefined for non-positive arguments.

In this exercise, the equation is solved by converting it into an equivalent exponential form, which can be solved using basic algebra.

Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. These are often encountered while dealing with logarithmic functions since logarithms are simply another way to express exponents.

To solve the given logarithmic equation, \( \log_4(3x+2) = 3 \), we convert it to its exponential form: \( 4^3 = 3x + 2 \). This transformation is key for solving the problem because it simplifies the equation into a form that is generally easier to work on.

Remember these points when dealing with exponential equations:

• Identify the base and the exponent. In this situation, 4 is the base and 3 is the exponent.
• On conversion, the formula is \( b^c = a \) where \( b \) is the base, \( c \) is the power, and \( a \) is the result.
• Once in exponential form, solve for the variable by isolating it with basic algebraic operations.

For our problem, solving \( 4^3 = 3x + 2 \) simplifies to \( 64 = 3x + 2 \), leading us to find the value of \( x \) step by step.

Domain of a Function

The domain of a function is the set of all possible input values (typically \( x \)-values) for which the function is defined. When dealing with logarithmic functions, it's crucial to check the domain because logarithms are only defined for positive arguments.

In the exercise, the original expression is \( \log_4(3x+2) \). This expression is only defined when \( 3x+2 > 0 \). Solving this inequality provides the domain for \( x \):

• Subtract 2 from both sides: \( 3x > -2 \)
• Divide by 3: \( x > -\frac{2}{3} \)

To see if the calculated \( x = \frac{62}{3} \) falls within this domain, check if \( \frac{62}{3} > -\frac{2}{3} \). Indeed, it does!This step is critical to confirm that the solution not only satisfies the converted and solved equation but also adheres to the constraints initially set by the logarithmic function. Hence, the checked value ensures that the mathematical process and result are valid and correct.