Problem 45 - Intermediate Algebra

Question

For Problems \(41-50\), solve each equation. $$ \log _{9} x=\frac{3}{2} $$

Step-by-Step Solution

Answer

The solution is \(x = 27\).

Step 1: Identify the Logarithm Base

The logarithm equation given is \( \log_{9} x = \frac{3}{2} \). The base here is \(9\). This means you're working with a base of 9 to unravel the logarithm.

Step 2: Convert the Logarithmic Equation to an Exponential Equation

Transform the logarithmic equation \( \log_{9} x = \frac{3}{2} \) into its equivalent exponential form, which states that: \(x = 9^{\frac{3}{2}} \).

Step 3: Calculate the Fractional Exponent

Recall that \(9^{\frac{3}{2}}\) means ether raising \(9\) to the power of \(\frac{3}{2}\) as follows: \(9^{\frac{1}{2}}\) gives the square root of \(9\), which is \(3\), and then cube this result: \(3^3 = 27 \).

Step 4: Verify the Solution

Verify that the solution \(x = 27\) satisfies the original logarithmic equation by checking: \(\log_{9} 27 = \frac{3}{2}\). Since \(9^{\frac{3}{2}} = 27\), the solution is correct.

Key Concepts

Logarithmic Equations

Logarithmic equations involve the use of logarithms to solve for an unknown variable. A logarithm is essentially the inverse operation of exponentiation, making it a powerful tool in mathematics. Since logarithms can transform multiplicative relationships into additive ones, they simplify complex calculations. In a logarithmic equation such as \( \log_{b} a = c \), the base \( b \) is the number that is raised to, what equals \( a \). So, solving the equation involves finding the value of \( a \) that fits this relationship.

To solve a logarithmic equation, first identify the base of the logarithm, as it guides the conversion to its exponential form. This transformation often simplifies the equation, making it easier to solve for the unknown.

• Convert logarithmic to exponential form.
• Solve the resulting expression to find the unknown variable.

Understanding logarithmic properties, like converting a log equation to its exponential form, is crucial in solving these types of problems.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is multiplied by itself as many times as indicated by the exponent. This operation is used to represent repeated multiplication in a concise way. For example, in the expression \( 9^{\frac{3}{2}} \), the base 9 is raised to the exponent \( \frac{3}{2} \).

In the context of solving exponential equations, understanding fractional exponents is essential. Fractional exponents, such as \( \frac{3}{2} \), imply two steps:

• First, take the root indicated by the denominator, \( \frac{1}{2} \) in this example means the square root.
• Second, raise the result to the numerator's power, 3 in this case, involves cubing.

Once you master these two steps, handling exponential expressions involving fractional powers becomes straightforward.

Problem-Solving

Problem-solving in mathematics is a systematic approach to finding solutions to equations. It involves understanding the problem, planning a strategy, executing that strategy, and finally checking the solution. For logarithmic equations, specific strategies are applicable.

When faced with an equation like \( \log _{9} x = \frac{3}{2} \), the steps to solve it effectively include:

• Identify the type of equation, e.g., logarithmic or exponential.
• Apply any transformations to convert the equation into a more solvable form, like turning a logarithmic equation into an exponential one.
• Carry out calculations carefully, paying attention to arithmetic rules.
• Verify your answer by plugging it back into the original equation.

This structured approach not only helps in finding the solution but also ensures that the answer is correct, building confidence in problem-solving skills.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, equations like the one discussed here operate on variables, unknowns which you solve for. Algebra provides the foundational techniques for equation solving, encompassing both linear and nonlinear equations.

A deep understanding of algebraic techniques is crucial for effective problem-solving, especially as equations become more complex:

• Use algebraic operations to isolate variables, simplify expressions, and solve equations systematically.
• Grasping concepts like logarithms and exponentials can lead to efficient solutions.

Algebra supports the transformation of equations, enabling you to approach solutions logically and systematically. Mastery of algebraic principles empowers you to tackle a wide range of mathematical challenges.