Problem 34 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \log _{8} x+\log _{8} x^{2}=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Apply Logarithm Property

Use the property \( \log_b (a) + \log_b (c) = \log_b (ac) \) to combine the logs: \[ \log_8(x) + \log_8(x^2) = \log_8(x \cdot x^2) = \log_8(x^3). \] Thus, the equation becomes: \[ \log_8(x^3) = 1. \]

Step 2: Convert Logarithm to Exponential Form

Convert the logarithmic equation \( \log_8(x^3) = 1 \) to its exponential form. This means that the base raised to the power of the logarithm equals the argument: \[ x^3 = 8^1. \] Thus we have: \[ x^3 = 8. \]

Step 3: Solve for x

To solve \( x^3 = 8 \), take the cube root of both sides: \[ x = \sqrt[3]{8}. \] Since the cube root of 8 is 2, we have: \[ x = 2. \]

Key Concepts

Logarithm Properties

Logarithms are powerful mathematical tools that reveal the relationship between multiplication and addition. A key property of logarithms, the product rule, states that the logarithm of a product is equal to the sum of logarithms of its factors. This is expressed as \( \log_b(a) + \log_b(c) = \log_b(ac) \). In our exercise, this property allowed us to combine \( \log_8(x) \) and \( \log_8(x^2) \) into \( \log_8(x^3) \).

• This simplification is useful for solving logarithmic equations.
• It converts the problem into a more manageable expression.
• By understanding and applying these properties, complex logarithmic problems become easier to tackle.

When dealing with logarithms, always check if these properties can be used to simplify expressions. They are crucial for reducing the complexity of the logarithmic terms.

Exponential Form

Exponential form is simply a different way of expressing mathematical equations involving powers. The relationship between logarithms and exponents is quite straightforward: if you have an equation \( \log_b(y) = x \), it can be rewritten as \( b^x = y \). This form is often more intuitive for solving equations because it allows us to focus on the base and power. In our problem, we used this conversion to transform \( \log_8(x^3) = 1 \) into \( x^3 = 8^1 \). This step:

• Highlights the simplicity of the exponential form.
• Emphasizes the direct link between the result and the base raised to a power.
• Makes the equation straightforward by eliminating logarithms early on.

Once in exponential form, you can proceed with algebraic techniques to solve for the variable of interest.

Cube Root

The cube root is a type of root where you try to find a number that, when multiplied by itself twice, gives the original number. It is written as \( \sqrt[3]{x} \) or \( x^{1/3} \). Solving for cube roots is an essential skill when dealing with equations like \( x^3 = 8 \). It involves finding the number (in this case, 2) that cubes to the original value.

• Cube roots can simplify expressions and solve polynomial equations.
• The process is straightforward when you recognize common cubed numbers (like 8, 27, 64, etc.).
• Understanding cube roots is helpful in various branches of mathematics beyond just algebra.

For example, recognizing that \( \sqrt[3]{8} = 2 \) quickly resolves the equation \( x^3 = 8 \) to \( x = 2 \). Mastery of cube roots streamlines problem-solving and helps you quickly identify solutions to cubic equations.