Problem 46 - College Algebra

Question

For the following exercises, solve each equation for \(x\). $$ \log _{2}(7 x+6)=3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{2}{7} \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{2}(7x + 6) = 3 \). This means that the logarithm with base 2 of the expression \(7x + 6\) is equal to 3.

Step 2: Convert Logarithm to Exponential Form

To solve the equation, convert the logarithmic equation to its equivalent exponential form. The property \( \log_b(a) = c \) can be rewritten as \( b^c = a \). Applying this property, we translate \( \log_{2}(7x + 6) = 3 \) to \( 2^3 = 7x + 6 \).

Step 3: Simplify the Exponential Expression

Calculate the value of the exponential expression. From the exponential form, \(2^3 = 8\), we have: \[ 8 = 7x + 6 \]

Step 4: Isolate the Variable x

Move the constant term from the right side to the left side by subtracting 6 from both sides of the equation:\[ 8 - 6 = 7x \] This simplifies to \( 2 = 7x \).

Step 5: Solve for x

To solve for \(x\), divide both sides of the equation by 7 to isolate it:\[ x = \frac{2}{7} \].

Key Concepts

Exponential Form

In the realm of mathematics, understanding how to convert a logarithmic equation to its exponential form is key to solving many problems. The exponential form helps us grasp what the logarithm actually represents.
To convert a logarithmic equation such as \( \log_b(a) = c \) into exponential form, you'll apply the rule: \( b^c = a \).

• "\( b \)" stands for the base of the logarithm.
• "\( a \)" represents the number for which the logarithm was taken.
• "\( c \)" is the logarithm of \( a \) to the base \( b \).

In the original exercise, the logarithmic equation is \( \log_{2}(7x + 6) = 3 \). By converting to exponential form, we transform it to the expression \( 2^3 = 7x + 6 \). This step simplifies the problem and makes it easier to handle because you're typically more familiar with solving exponential equations than those in logarithmic form.

Logarithmic Properties

Logarithms have the unique ability to transform multiplication into addition and division into subtraction, making complex calculations simpler. Knowing these properties can be incredibly useful.
There are several key logarithmic properties that are often used:

• Product Rule: \( \log_b(MN) = \log_bM + \log_bN \)
• Quotient Rule: \( \log_b\left( \frac{M}{N} \right) = \log_bM - \log_bN \)
• Power Rule: \( \log_b(M^p) = p\cdot\log_bM \)
• Change of Base Formula: \( \log_b a = \frac{\log_c a}{\log_c b} \)

In this specific exercise, we did not utilize these properties directly, but understanding them is crucial for dealing with more advanced logarithmic equations and can often simplify the process significantly. In particular, knowing how to manipulate and transform logarithmic expressions can make finding solutions far more intuitive.

Solving Equations

Solving equations, whether they are linear, quadratic, or logarithmic, involves finding the value of the variable that makes the equation true. Let's discuss the process using the given exercise as an example, which is \( \log_{2}(7x + 6) = 3 \). First, we converted the logarithmic equation to exponential form, giving us \( 2^3 = 7x + 6 \). From here, solving the equation means isolating the unknown variable \( x \).
To isolate \( x \), start by performing operations that simplify one side of the equation. In the expression \( 8 = 7x + 6 \), subtract 6 from both sides to yield \( 2 = 7x \). This step effectively removes the constant term on the right side, leaving \( 7x \) alone. Next, divide both sides by 7 to solve for \( x \):\[ x = \frac{2}{7} \]Ultimately, solving the equation involves transforming the equation step-by-step, adhering to arithmetic rules as you aim to isolate the variable. Equations can vary greatly in complexity, but the fundamental principles of isolating variables and simplifying expressions are consistently applied.