Problem 46 - College Algebra with Corequisite Support

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 Given Equation

The given equation is \( \log_{2}(7x+6) = 3 \). A logarithmic equation needs to be converted to its exponential form to solve for \( x \).

Step 2: Convert Logarithmic Form to Exponential Form

Use the definition of a logarithm to rewrite the equation in exponential form. The equation \( \log_{2}(7x+6) = 3 \) is equivalent to the exponential equation \( 2^3 = 7x + 6 \).

Step 3: Solve the Exponential Equation

Calculate \( 2^3 \) which equals \( 8 \). This transforms the equation to \( 8 = 7x + 6 \).

Step 4: Isolate the Variable

Subtract \( 6 \) from both sides to isolate terms with \( x \): \( 8 - 6 = 7x \). This simplifies to \( 2 = 7x \).

Step 5: Solve for x

Divide both sides by \( 7 \) to solve for \( x \): \( x = \frac{2}{7} \).

Key Concepts

Exponential Form

Understanding exponential form is key when solving logarithmic equations. In mathematics, converting from logarithmic form to exponential form lets us express equations involving exponential relationships. If you have a logarithmic expression like \( \log_b(a) = c \), its equivalent exponential form is \( b^c = a \). This transformation is useful because it simplifies the process of solving for the unknown variable.In the exercise \( \log_{2}(7x+6) = 3 \), converting to exponential form involves identifying:

• Base \( b = 2 \)
• Result \( a = 7x + 6 \)
• Exponent \( c = 3 \)

Thus, the equivalent exponential expression becomes \( 2^3 = 7x + 6 \). This representation makes further steps like solving for \( x \) more straightforward.

Logarithmic Functions

Logarithms are the inverse of exponents and are a fundamental tool in mathematics for solving equations involving exponential growth or decay. A logarithmic function is of the form \( y = \log_b(x) \), where:

• \( y \) is the logarithm of \( x \)
• \( b \) is the base of the logarithm
• \( x \) is the argument or result of the operation

Using logarithmic functions, one can determine the power to which a base must be raised to produce a given number. It's crucial to grasp this concept when dealing with equations.In our given problem, \( \log_{2}(7x+6) = 3 \), we're asked what power \( 2 \) must be raised to with the result \( 7x + 6 \). The answer here is \( 3 \), but knowing how to interpret and manipulate logarithmic functions enables a deeper understanding of how to solve such equations effectively.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to isolate the variable of interest. This process often includes operations like addition, subtraction, multiplication, division, and sometimes factoring or expanding terms. First, you convert the logarithmic equation into an exponential equation, as seen with the conversion \( 2^3 = 7x + 6 \).Next, simplifying this exponential equation requires standard algebraic steps:

• Calculate the exponential value: \( 2^3 = 8 \)
• Isolate the term containing \( x \) by subtracting \( 6 \) from both sides, leading to \( 8 - 6 = 7x \)
• Resulting in \( 2 = 7x \)
• Finally, divide both sides by \( 7 \) to solve for \( x \): \( x = \frac{2}{7} \)

These steps showcase how algebraic manipulation simplifies and solves equations, making complex problems more approachable and solvable, enhancing comprehension of such mathematical processes.