Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{18}(x)=2\)

Step-by-Step Solution

Answer

The value of \( x \) is 324.

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{18}(x) = 2 \). This means we are looking for a number \( x \) such that when the base 18 is raised to a power, we get \( x \).

Step 2: Convert to Exponential Form

To convert the logarithmic equation to its exponential form, use the identity \( \log_b(a) = c \) implies \( b^c = a \). Here, \( b = 18 \), \( a = x \), and \( c = 2 \). Therefore, the equation becomes: \( 18^2 = x \).

Step 3: Simplify the Exponential Expression

Calculate \( 18^2 \) to find \( x \). The calculation is: \( 18^2 = 18 \times 18 = 324 \).

Step 4: State the Solution

The value of \( x \) that satisfies the original logarithmic equation is \( x = 324 \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a fundamental aspect of mathematics, particularly when dealing with exponential equations. A logarithm answers the question: "To what exponent must the base be raised, to get a certain number?"
For instance, in the expression \( \log_{b}(a) = c \), \( b \) is the base, \( a \) is the argument, and \( c \) is the power or exponent.
Essentially, this means that raising the base \( b \) to the power of \( c \) equals \( a \). Logarithms are the inverses of exponential functions and they play a vital role in simplifying expressions, especially when solving equations involving exponential growth or decay.

• Logarithmic expressions help in solving equations where the variable is an exponent.
• They simplify complex exponential calculations into more manageable forms.

Understanding logarithms can make dealing with large numbers or unknown exponents much simpler, as they often simplify to integer forms in basic applications.

Exponential Form

The exponential form is a way of representing numbers using a base and an exponent. It is the inverse of taking a logarithm.
In exponential terms, if we have a logarithmic expression like \( \log_{18}(x) = 2 \), the equivalent exponential form is written as \( 18^2 = x \).
Here, the base is 18, and the exponent is 2.

• Base: The number that is being multiplied by itself.
• Exponent: The power to which the base is raised, indicating how many times it is multiplied by itself.

Using exponential form is crucial for solving logarithmic equations because it converts a potentially complex logarithmic expression into a straightforward multiplication problem. This makes finding the value for \( x \) much simpler by direct calculation.

Solving Equations

Solving equations, particularly those involving logarithms, often requires converting logarithmic expressions to exponential form. This involves a step-by-step approach:
Start by identifying the components of the logarithmic equation, such as the base, the argument, and the result.
Next, rewrite the logarithmic equation in its exponential counterpart, using the property that \( \log_b(a) = c \) translates to \( b^c = a \).

• Convert the logarithmic equation into exponential form.
• Solve for the unknown variable by performing the necessary calculations.
• Check your solution to ensure it satisfies the original equation.

This method helps in breaking down the equation into more manageable parts, leading to quick and accurate solutions.

Mathematical Calculations

Mathematical calculations are at the heart of solving equations. Once a logarithmic equation is transformed into its exponential form, the primary task is to execute simple arithmetic operations.
For example, in the equation \( 18^2 = x \), we multiply 18 by itself to find \( x \).
The calculations are straightforward but critical, as accuracy in this step determines the correctness of the solution.

• Perform arithmetic confidently and accurately.
• Use logical verification to double-check your work.
• Understanding the arithmetic ensures reliable problem-solving.

These calculations follow mathematical principles that ensure precision. Mastery of basic arithmetic leads to ease in handling algebraic concepts, making solving such equations efficient and effective.