Problem 94 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log _{3} 5 x=10$$

Step-by-Step Solution

Answer

\( x = 11809.8 \)

Step 1: Understand the equation

The equation given is \( \log_{3}(5x) = 10 \). This is a logarithmic equation with base 3.

Step 2: Convert to exponential form

To solve the equation, we first convert the logarithmic form to an exponential form. According to the properties of logarithms, if \( \log_{b}(a) = c \), then \( a = b^{c} \). Applying this rule here gives us: \( 5x = 3^{10} \).

Step 3: Calculate the exponential value

Calculate \( 3^{10} \) to find the exponential value. \( 3^{10} \) equals 59049.

Step 4: Solve for \( x \)

After calculating the exponential expression, we have \( 5x = 59049 \). We then solve for \( x \) by dividing both sides by 5: \( x = \frac{59049}{5} \).

Step 5: Compute the division

Divide 59049 by 5 to find the value of \( x \). Performing the calculation gives \( x = 11809.8 \).

Step 6: Check the solution

To ensure that your solution is correct, substitute \( x = 11809.8 \) back into the original equation and verify that \( \log_{3}(5 \times 11809.8) \approx 10 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve the logarithm of a variable or expression. In such equations, the unknown quantity is usually part of the logarithmic expression, as seen in an equation like \( \log_b(x) = y \). Here, \( b \) is the base of the logarithm, \( x \) is the argument or the value we are taking the logarithm of, and \( y \) is the result.

To solve a logarithmic equation, we often have to manipulate it in such a way that the unknown can be isolated. This can involve using the properties of logarithms or converting the equation into a different form. The key here is to understand what the logarithm represents, which is the power to which the base must be raised to produce the given number. Logarithmic equations are highly useful in many fields like engineering, computer science, and finance because they can simplify problems involving exponential relationships.

For example, the given logarithmic equation \( \log_{3}(5x) = 10 \) requires us to understand the role of the base '3' and how the expression '5x' is the argument being evaluated by the logarithm.

Exponential Form Conversion

Converting logarithmic equations to exponential form is a crucial step in solving them. The essence of a logarithm equation \( \log_b(a) = c \) is that it can be restated in its equivalent exponential form \( a = b^c \). This conversion is based on the definition of logarithms, allowing us to transition from a logarithmic form to a more manageable form where traditional algebraic methods can be applied.

In the example equation \( \log_{3}(5x) = 10 \), converting to exponential form results in \( 5x = 3^{10} \). This step requires recognizing that the base of the logarithm becomes the base of the exponential expression, and the value on the other side of the logarithmic equation becomes the exponent.

• The logarithmic base '3' becomes the base in the exponential equation.
• The result '10' is used as the power or exponent.

By transforming a logarithmic problem into an exponential one, it becomes possible to solve for \( x \) using straightforward algebra. This is particularly useful because exponential functions grow quickly with increasing exponents, and with calculators, computing powers of numbers is generally simpler.

Properties of Logarithms

Understanding the properties of logarithms is essential for manipulating and solving logarithmic equations. These properties give us tools to simplify logarithms, expand them, or combine them, which can be very handy when solving equations or simplifying expressions involving logarithms.

Some of the primary properties of logarithms include:

• The product property: \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• The quotient property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• The power property: \( \log_b(m^n) = n\log_b(m) \).
• The change of base formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), where \( k \) is a new base.

Applying these properties can simplify an equation or an expression by rearranging or breaking down logarithms into simpler parts. For instance, if the logarithmic equation involved a product such as \( \log_{3}(5x) = \log_{3}(5) + \log_{3}(x) \), this could be used to separate terms and make it easier to solve. The properties of logarithms make them versatile and accessible for mathematical manipulation, facilitating the solution of complex logarithmic equations.