Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Give the exact answer. $$\log _{2}(x+1)=3$$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Understand the Logarithm

The equation given is a logarithmic equation, which means it involves a logarithm. In this case, the logarithm is base 2 and is expressed as \(\log_{2}(x+1) = 3\). This implies that \(x + 1\) is the quantity in question, and \(3\) is the result of the logarithm.

Step 2: Rewrite the Equation in Exponential Form

A logarithmic equation can be rewritten in an exponential form. Since \( \log_{2}(x+1) = 3 \), this can be rewritten as an exponential equation: \(x+1 = 2^3\). This transformation is done because a logarithm tells us the exponent that the base (2 in this case) must be raised to in order to get the number inside the logarithm (\(x+1\)).

Step 3: Calculate the Exponential

Calculate \(2^3\). This means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\). Thus, the equation \(x + 1 = 8\) is obtained.

Step 4: Isolate x

To find the value of \(x\), solve the equation \(x + 1 = 8\). Subtract 1 from both sides of the equation to isolate \(x\): \(x = 8 - 1\).

Step 5: Simplify the Expression

After subtracting, the equation becomes \(x = 7\). Ensure this is the simplest form of the solution and check if it satisfies the original logarithmic equation. Substitute back to confirm: \(\log_{2}(7 + 1) = \log_{2}(8)\), which equals 3. Hence, the solution is verified.

Key Concepts

Exponential Form

Logarithms and exponential functions are deeply connected. Understanding how to switch between the two forms is essential in solving logarithmic equations. When we talk about rewriting a logarithmic equation such as \(\log_{b}(y) = x\) into an exponential form, we mean expressing it as \(y = b^x\).
In our example equation, \(\log_{2}(x+1) = 3\), we begin by identifying the base of the logarithm, which is 2. Then, we express the equation in exponential form: \(x+1 = 2^3\). This tells us that to get the number inside the logarithm (here \(x+1\)), we raise the base 2 to the power of 3.
This conversion is crucial because exponential equations are often easier to manipulate and solve than logarithmic ones. By expressing our problem in exponential form, we can proceed with straightforward mathematical operations to find a solution.

Base of Logarithm

The base of the logarithm is foundational in understanding logarithmic expressions. In our equation \(\log_{2}(x+1) = 3\), the base of the logarithm is 2. This base indicates the number that needs to be multiplied by itself a specific number of times—indicated by the logarithm's output—to result in the value inside the logarithm.
Here's how it works in practice:

• The equation \(\log_{2}(x+1) = 3\) specifies that we have to find a power of 2 (the base) that equals the quantity \(x+1\).
• Rewriting this using the base as referenced, it becomes \(x+1 = 2^3\).
• Understanding this base concept lets us restructure the logarithmic equation into a simpler exponential format.

The base of the logarithm plays a central role and recognizing it helps unravel the expression and solve the variable effectively.

Isolation of Variable

Isolating the variable is a fundamental step in solving equations. With the equation in its exponential form \(x+1 = 8\), the task now is to isolate \(x\) to determine its value independently.
Here's a simple way to do this:

• Start from the equation obtained after converting to exponential form: \(x+1 = 8\).
• Subtract 1 from both sides: \(x = 8 - 1\).
• This simplification leads to \(x = 7\), fully isolating the variable and solving for its value.

Isolating the variable often involves performing basic arithmetic operations like addition or subtraction. It's a step meant to express the variable by itself on one side of the equation, making it straightforward to identify and verify the solution. This isolation allows us to easily check our results by substituting back into the original problem to ensure correctness.