Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{4}\left(x^{3}+37\right)=3$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Understanding the Logarithmic Equation

The given equation is \( \log_{4}(x^3 + 37) = 3 \). This means that the logarithm base 4 of \( x^3 + 37 \) equals 3. According to the definition of logarithms, this implies that \( 4^3 = x^3 + 37 \). Our goal is to solve for \( x \).

Step 2: Exponentiating the Expression

Since \( \log_{4}(x^3 + 37) = 3 \) implies \( x^3 + 37 = 4^3 \), we need to calculate \( 4^3 \). Calculate this to find that \( 4^3 = 64 \). Thus, the equation becomes \( x^3 + 37 = 64 \).

Step 3: Isolating the Variable Term

To isolate \( x^3 \), subtract 37 from both sides:\[ x^3 + 37 - 37 = 64 - 37 \]\[ x^3 = 27 \].

Step 4: Solving for x

Now, solve for \( x \) by finding the cube root of both sides of the equation:\[ x = \sqrt[3]{27} = 3 \].

Step 5: Verifying the Solution

Confirm by substituting \( x = 3 \) back into the original equation:\[ \log_{4}(3^3 + 37) = \log_{4}(27 + 37) = \log_{4}(64) \].Since \( 64 = 4^3 \), \( \log_{4}(64) = 3 \) which matches our original equation, confirming that \( x = 3 \) is correct.

Key Concepts

Exponents

Exponents are a mathematical way to express repeated multiplication of a number by itself. For instance, in our equation, we have an expression like \( 4^3 \). This means \( 4 \) is multiplied by itself three times, resulting in 64. Exponents are common in algebra and are crucial for solving many types of equations, including logarithmic ones.

• In an expression \( a^b \), \( a \) is the base and \( b \) is the exponent.
• The result shows how many times the base is multiplied by itself.
• Exponents follow specific rules, like \( a^0 = 1 \) (for any non-zero \( a \)) and \( a^1 = a \).

The exponent rule is also a foundational concept when solving logarithmic equations, like in our task. Great understanding of exponents will help in smoothly transitioning to logarithms, as they are inherently connected.

Cube Roots

Cube roots are the inverse operation of cubing a number. When you take the cube root of a number, you're finding a value that, when multiplied by itself twice, equals the original number. For instance, the cube root of 27 is 3, since \( 3 \times 3 \times 3 = 27 \).

• The cube root of a number \( a \) is denoted as \( \sqrt[3]{a} \).
• Cube roots help solve equations where a variable is cubed, as seen in \( x^3 = 27 \).

In our solution, finding the cube root of 27 was an essential step to solve for \( x \). Understanding cube roots helps in reversing the effect of cubing a number, a common requirement in algebra.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They help to solve equations where the variable is an exponent, like in \( \log_4(x^3 + 37) = 3 \). The function tells us the power we must raise the base to achieve a specific number.

• In \( \log_b(a) = c \), \( b^c = a \) shows the link between logarithms and exponents.
• Logarithms have distinctive properties like \( \log_b(mn) = \log_b(m) + \log_b(n) \) and \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• They are used in many fields, from science to engineering, for calculations involving exponential growth or decay.

By converting a logarithmic equation into an exponential form, as seen in the solution \( x^3 + 37 = 4^3 \), we often make complex problems more manageable. Understanding logarithmic functions is key to solving such equations and is a fundamental concept that interplays with exponents frequently.