Problem 35 - Algebra and Trigonometry

Question

Use the definition of the logarithmic function to find \(x\). $$ \text { (a) } \log _{x} 16=4 \quad \text { (b) } \log _{x} 8=\frac{3}{2} $$

Step-by-Step Solution

Answer

For (a), \( x = 2 \); for (b), \( x = 4 \).

Step 1: Understand the Logarithmic Equation

Logarithms express the relationship between numbers in the form of exponential equations. The equation \( \log_{b}(a) = c \) is equivalent to the exponential form \( b^c = a \). Identify each element from the problem statements:(a) \( \log _{x} 16 = 4 \) means \( x^4 = 16 \).(b) \( \log _{x} 8 = \frac{3}{2} \) means \( x^{\frac{3}{2}} = 8 \).

Step 2: Solve for \(x\) in Equation (a)

For Equation (a), \( x^4 = 16 \). To solve for \( x \), take the fourth root of both sides:\[ x = \sqrt[4]{16} = \sqrt[4]{2^4} = 2 \] Thus, \( x = 2 \).

Step 3: Solve for \(x\) in Equation (b)

For Equation (b), \( x^{\frac{3}{2}} = 8 \). To solve for \( x \), first square both sides to rid the fraction:\[ (x^{\frac{3}{2}})^2 = 8^2 \]This simplifies to \( x^3 = 64 \). Take the cube root of both sides to solve for \( x \):\[ x = \sqrt[3]{64} = \sqrt[3]{4^3} = 4 \]Hence, \( x = 4 \).

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. For example, the equation \( x^4 = 16 \) is an exponential equation because the variable \( x \) is an exponent.

• These equations often arise from logarithmic functions since a logarithm represents an exponent.
• The key understanding here is that any equation of the form \( a^x = b \) can be solved by finding the appropriate root or logarithm of \( b \).

To effectively handle exponential equations:
Understand the base and the result. Here, you know \( x^4 \) results in 16, which indicates that 16 is the base raised to the fourth power.
Remember that solving these equations generally involves finding the value of the base that, when raised to the given power, equals the number provided on the other side of the equation.
In the example, determining the value of \( x \) involves taking the fourth root of both sides, leading us to \( x = 2 \).

Solving Logarithmic Equations

A logarithmic equation involves a logarithm with a variable inside. It generally takes the form \( \log_b(a) = c \), which ends up representing the equation in exponential form \( b^c = a \). This representative transformation is key to solving logarithmic equations.

The two core steps when solving such equations:

• Convert the logarithmic equation into its corresponding exponential form.
• Solve the resulting exponential equation.

For instance, in the problem \( \log_{x} 16 = 4 \), converting this to exponential form gives us the equation \( x^4 = 16 \). Solving this, as previously discussed, leads to \( x = 2 \).

In the case of \( \log_{x} 8 = \frac{3}{2} \), the exponential form becomes \( x^{\frac{3}{2}} = 8 \). To solve, square both sides resulting in \( x^3 = 64 \), and then find the cube root, leading to \( x = 4 \).

Roots of Equations

Roots of equations occur when solving polynomial equations. They are essentially values that satisfy a given polynomial equation.

• The root of an equation \( x^n = a \) determines the value of \( x \) that makes the equation true.
• This involves taking the appropriate root (like square root, cube root, etc.) based on the exponent in the equation.

For example, the equation \( x^4 = 16 \) necessitates finding the fourth root of 16, hence \( x = \sqrt[4]{16} = 2 \).

Similarly, in \( x^3 = 64 \), solving for \( x \) involves finding the cube root of 64, resulting in \( x = 4 \).

Understanding roots involves recognizing patterns in numbers and being familiar with powers and their corresponding roots. This knowledge greatly simplifies finding solutions to both logarithmic and exponential equations.