Problem 49 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$3 \log _{2}\left(3 x^{2}+2\right)+1=2$$

Step-by-Step Solution

Answer

Solutions are \( x = \pm \sqrt{\frac{2^{1/3} - 2}{3}} \).

Step 1: Isolate the logarithmic term

Start by isolating the logarithmic term in the equation. Subtract 1 from both sides:\[ 3 \log_{2}(3x^2 + 2) = 2 - 1 \]which simplifies to:\[ 3 \log_{2}(3x^2 + 2) = 1 \].

Step 2: Divide by the coefficient of the logarithm

Divide both sides of the equation by 3 to further isolate the logarithmic expression:\[ \log_{2}(3x^2 + 2) = \frac{1}{3} \].

Step 3: Convert the logarithmic equation to exponential form

Use the property of logarithms that states \( \log_{b}(a) = c \) is equivalent to \( a = b^c \). Apply this to the equation:\[ 3x^2 + 2 = 2^{rac{1}{3}} \].

Step 4: Solve for \( x^2 \)

Subtract 2 from both sides:\[ 3x^2 = 2^{\frac{1}{3}} - 2 \]Divide by 3:\[ x^2 = \frac{2^{\frac{1}{3}} - 2}{3} \].

Step 5: Solve for \( x \)

Take the square root of both sides, remembering to consider both positive and negative roots:\[ x = \pm \sqrt{\frac{2^{\frac{1}{3}} - 2}{3}} \].

Step 6: Calculate using a calculator

Use a calculator to evaluate \(2^{\frac{1}{3}}\) and find the decimal approximation of the solutions. Ensure to check if both solutions satisfy the original equation and are valid.

Key Concepts

Logarithms

Logarithms are a powerful mathematical tool that allows us to solve equations involving exponents. It is essentially the inverse operation of exponentiation, which means it helps us find the exponent when the base and the result are known. In simpler terms, if you know that a certain number raised to a power results in another number, the logarithm helps you figure out what that power is.

• A logarithm is denoted as \(\log_{b}(a) = c\), where \(b\) is the base, \(a\) is the number you want the log of, and \(c\) is the power that \(b\) must be raised to get \(a\).
• For example, \(\log_{10}(1000) = 3\) because \(10^3 = 1000\).

Logs are particularly useful in various fields such as science and engineering where they simplify calculations involving large or small numbers. Understanding how to manipulate logarithms is key to solving logarithmic equations.

Exponents

Exponents represent repeated multiplication. When you see a number raised to a power, it tells you how many times to multiply the base by itself. For instance, in \(2^3\), 2 is multiplied by itself 3 times, resulting in 8. Exponents can be whole numbers, fractions, or even negative numbers.

• Fractional exponents indicate roots, so \(2^{1/3}\) means the cube root of 2.
• Negative exponents indicate reciprocals, so \(2^{-3} = 1/2^3 = 1/8\).

Understanding how exponents work is crucial for converting logarithmic equations into exponential form, which is a common technique used to solve them. This knowledge helps in transforming complex equations into simpler, solvable forms.

Equation Solving

Equation solving involves finding the values of unknowns that make the equation true. For logarithmic equations, this often means manipulating the equation to isolate the logarithmic part and then converting it to an exponential form. **Key Steps in Solving Equations:**

• Isolate the variable or term you’re solving for. This often involves moving terms to one side of the equation.


• If dealing with a logarithm, convert it to exponential form using \(\log_{b}(a) = c\) implies \(a = b^c\).


• Perform necessary actions to solve for the unknown, such as subtracting, dividing, or taking roots.

These steps allow you to simplify the equation step-by-step, making it easier to find the solution, as seen in our specific logarithmic equation example.

Mathematical Expressions

Mathematical expressions consist of numbers and symbols grouped together to represent calculations. They can include operations like addition, subtraction, multiplication, division, as well as more complex operations like exponents and logarithms.**Components of Mathematical Expressions:**

• Constants and Variables: Numbers and letters that form the backbone of expressions.


• Operators: Signs like \(+\), \(-\), \(\times\), \(\div\) that denote the operations to perform.


• Functions: Such as \(\log\) or \(\sin\), indicating particular operations to perform on variables or numbers.

In the context of solving equations, particularly logarithmic ones, understanding how to manipulate these expressions and simplify them while maintaining equality is essential. This allows you to follow through with the necessary transformations to isolate variables and ultimately find solutions.