Problem 53 - Algebra 2 and Trigonometry

Question

Solve each equation for the variable. \(\log _{2} 8+\log _{3} 9=\log _{b} 100,000\)

Step-by-Step Solution

Answer

The value of \(b\) is 10.

Step 1: Apply Logarithmic Properties

Start by recognizing the logarithmic property that states when you add two logarithms with different bases, they can be expressed differently. In this case, use the property that allows you to convert logs to a common base or express each log in exponential form: \(\log_2 8 = 3 \) since \(2^3 = 8\) and \(\log_3 9 = 2 \) since \(3^2 = 9\).

Step 2: Calculate the Sum of Known Logarithms

Add the values obtained from the first step:\(\log_2 8 + \log_3 9 = 3 + 2 = 5\).

Step 3: Express in Terms of Base b

Now express the right side of the equation in terms of \(\log_b\):\(\log_b 100,000\). Recall that \(100,000 = 10^5\), which can be expressed in logarithmic terms as:\(\log_b 100,000 = 5 \log_b 10\).

Step 4: Equate and Solve for b

Set the logarithms equal to each other from both sides of the equation:\(5 = 5 \log_b 10\).To isolate \(\log_b 10\), divide both sides by 5:\(1 = \log_b 10\).

Step 5: Convert Logarithm to Exponential Form

Use the definition of logarithms to convert this equation to exponential form:\(b^1 = 10\).Thus, \(b = 10\).

Key Concepts

Understanding Logarithmic Properties

Logarithmic properties are the rules that govern how logarithms behave. These properties help simplify logarithmic expressions and solve logarithmic equations. A key property used in this exercise is the change of base and the conversion of logarithmic functions through known equivalent exponential forms.
For instance:

• The property \ \( \log_b(x^n) = n \cdot \log_b(x) \ \) allows us to manipulate logarithmic expressions.
• The equation \ \( \log_a(b) + \log_a(c) = \log_a(b \cdot c) \ \) demonstrates the property that the sum of logarithms of the same base is equivalent to the logarithm of the product of their arguments.

Recognizing these properties enables us to simplify complex expressions and solve equations that contain logarithms.

Logarithmic Conversion Techniques

Converting a logarithm means rewriting it in a different form without changing its value. This can help solve equations or compare logarithms with different bases.
For example:

• Convert \ \( \log_2 8 \ \) to the exponential form: Since \( 2^3 = 8, \) we can say \ \( \log_2 8 = 3 \ \).
• Similarly, \ \( \log_3 9 = 2 \ \) because \ \( 3^2 = 9 \ \).

By converting the logarithms into integers, solving the actual equation becomes much more straightforward. This conversion is particularly useful for choosing a suitable base in logarithmic equations or synching different logarithmic expressions to a common base.

Steps in Solving Logarithmic Equations

Solving logarithmic equations involves finding the value of the unknown variable for which the equation holds true. Here’s a simplified breakdown of how to approach these equations:

• First, convert each logarithm to its basic exponential form, if possible.
• Combine logarithms using properties and simplify to generalize their expression.
• Calculate known quantities to simplify the equations further. This step often involves conversions to practical numbers, like in the problem at hand where \(\log_2 8 = 3\qquad \text{and}\qquad \log_3 9 = 2\).
• Finally, set up the simplified forms equal and solve for the variable, ensuring all steps adhere to basic logarithmic properties.

Focusing on each of these steps individually helps navigate through complex expressions and find precise solutions.

Converting Logarithms to Exponential Form

The exponential form is a way to express logarithms as powers of their bases. This method simplifies understanding and solving logarithmic equations.

• In our example : \(\log_b(100,000) = 5\log_b(10)\) simplifies via conversion to exponential, which helps set equations equal and solve for \(b\).
• We know that if \(\log_b(x) = y\), then in exponential form \(b^y = x\).

This method can effectively solve for unknown variables by isolating them in easier-to-manage exponential expressions. This conversion also helps in checking a solved equation for accuracy. It's the fundamental ground of many algebraic manipulations, making it crucial for mastering logarithmic equations.