Problem 93 - Intermediate Algebra

Question

For Problems \(81-97\), solve each of the equations. $$ \log _{2} 5+\log _{2}(x+6)=3 $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{22}{5} \).

Step 1: Understand Logarithmic Properties

Recall that when two logarithms with the same base are added, you can multiply their arguments. This property can be stated as \( \log_b A + \log_b B = \log_b (A \times B) \). In this equation, we have: \( \log_2 5 + \log_2(x+6) = \log_2 [5(x+6)] \).

Step 2: Apply the Property to the Given Equation

Using the logarithmic property from Step 1, rewrite the equation: \( \log_2 [5(x+6)] = 3 \).

Step 3: Convert the Logarithmic Equation to an Exponential Equation

Recall that \( \log_2 A = B \) is equivalent to \( 2^B = A \). Apply this to the equation: \( 5(x+6) = 2^3 \).

Step 4: Solve the Exponential Equation

Calculate \( 2^3 \), which equals 8. So the equation becomes \( 5(x+6) = 8 \).

Step 5: Simplify the Equation

Divide both sides by 5 to isolate \( x+6 \):\( x+6 = \frac{8}{5} \).

Step 6: Solve for x

Subtract 6 from both sides to solve for \( x \): \( x = \frac{8}{5} - 6 \). Convert \( 6 \) to \( \frac{30}{5} \) so both numbers have the same denominator: \( \frac{8}{5} - \frac{30}{5} = -\frac{22}{5} \). Thus, \( x = -\frac{22}{5} \).

Key Concepts

Logarithmic Equations

Logarithmic equations like \( \log_{2} 5 + \log_{2}(x+6) = 3 \) often appear challenging at first. Their power lies in how they transform complex multiplication problems into easier addition problems.
To understand logarithmic equations, it's essential to grasp how logarithms work. When two logs with the same base are added, they can be combined into a single log by multiplying their arguments. This property is expressed as \( \log_b A + \log_b B = \log_b (A \times B) \).
In simple terms, solving logarithmic equations involves two steps:

• Combine or simplify the logarithmic terms using properties.
• Convert the logarithmic equation to an exponential form to solve for the variable.

Logarithmic Properties

Understanding logarithmic properties is key to solving equations effectively. One fundamental property of logarithms is how they interact with multiplication and division. For example:

• \( \log_b (A \times B) = \log_b A + \log_b B \)
• \( \log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B \)

These properties allow us to rewrite equations, making them easier to handle. In our specific problem, we apply \( \log_b A + \log_b B = \log_b (A \times B) \) to rewrite the two logs as one: \( \log_2 [5(x+6)] \).
Mastery of these properties is vital, as it simplifies the equation and leads to a shortcut that makes solving them much more straightforward.

Exponential Equations

Once you've simplified the logarithmic equation using its properties, the next step involves converting it into an exponential equation. This conversion demystifies the unknown variable. Recall that if \( \log_b A = C \), then \( b^C = A \).
This same principle is applied to convert \( \log_2 [5(x+6)] = 3 \) into the form \( 5(x+6) = 2^3 \).
Solving this exponential equation is typically straightforward; you calculate the power and isolate the variable. In the context of our equation:

• Calculate \( 2^3 \) which gives 8.
• Substitute this value back into the equation.
• Isolate \( x \) by simple arithmetic operations.

Converting log equations to exponentials effectively opens doors to simpler solutions.