Problem 31 - Calculus of a Single Variable

Question

Solving an Equation In Exercises \(25-34,\) solve the equation accurate to three decimal places. $$ \log _{2}(x-1)=5 $$

Step-by-Step Solution

Answer

The solution to the equation \(\log_{2}(x-1)=5 \) is \( x=33 \)

Step 1 - Convert Log Equation to Exponential Equation

First, let's change the log equation to a simpler form which is an exponential equation. A logarithmic equation \( \log_{b}(a)=c \) can be converted to the exponential form \( b^c=a \). Using this property, the given equation \(\log_{2}(x-1)=5 \) can be transformed into \( 2^5=x-1 \)

Step 2 - Solve for x

Now, using the new simpler form of equation, let's solve for x. If we add 1 to both sides of the equation \( 2^5=x-1 \) it becomes \( 2^5+1=x \)), which simplifies to \( 32+1=x \) or \( x = 33 \).

Step 3 - Check if the Solution is Correct

By inserting the value \( x=33 \) in the original equation \( \log_{2}(x-1)=5 \), it gives us \( \log_{2}(33-1)=5 \) or \( \log_{2}(32)=5 \). Since the logarithm base 2 of 32 equals to 5, the solution is correct.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms. A basic logarithm equation is expressed in the form \( \log_b(a) = c \), where \( b \) is the base, \( a \) is the value, and \( c \) is the exponent. These types of equations are useful in various fields like science and engineering for solving problems related to exponential growth or decay.

Understanding how to manipulate logarithmic equations can help simplify complex problems. The essential skill here is to be able to convert these equations into a more manageable form, such as exponential equations.

• Base: The number that is being raised to a power.
• Exponent: Indicates how many times the base is multiplied by itself.
• Value: The result of the base raised to the exponent.

A key thing to remember with logarithmic equations is the relationship between logs and exponentials which can simplify solving them.

Mathematical Problem Solving

Mathematical problem solving often involves breaking down complex equations into simple, manageable parts. This approach requires a systematic method or strategy to reach a solution. Here’s a simple workflow for solving equations similar to the ‘logarithmic to exponential’ type:

• Identify the type of equation: Recognizing whether it’s a log or exponential form is crucial.
• Conversion: Convert the equation to a simpler form, often from logarithmic to exponential.
• Solve the simpler equation: Once converted, solving often becomes straightforward.
• Verify the solution: Substitute the obtained solution back into the original equation to confirm its correctness.

Being methodical and patient, and following these steps, helps avoid errors. It ensures accuracy, especially when answering problems that require a precise answer, such as the one solved to three decimal places.

Equation Conversion

Equation conversion refers to transforming an equation from one format to another to simplify the process of solving it. In many mathematical problems, particularly involving logarithms, being able to convert equations can be a powerful tool. For instance, logarithmic equations can often be converted into exponential equations using the equivalence \( \log_{b}(a)=c \) converts to \( b^c=a \).

This conversion allows for easier manipulation of the equation, eventually making the calculation simpler. For example, in the exercise provided, converting \( \log_{2}(x-1)=5 \) into exponential form \( 2^5=x-1 \) allows us to solve for \( x \) easily.

• Simplification: Converts complex forms like logs into straightforward exponential forms.
• Comprehension: Provides a clearer view of the relationships between numbers.
• Efficiency: Simplifies complex mathematical operations and reduces computation time.

Mastering equation conversions is essential for students aiming to excel in algebra and calculus.