Problem 34 - College Algebra

Question

For the following exercises, use the definition of a logarithm to solve the equation. $$ 2 \log (8 n+4)+6=10 $$

Step-by-Step Solution

Answer

The solution is \( n = 12 \).

Step 1: Isolate the Logarithmic Term

Subtract 6 from both sides of the equation to isolate the logarithmic term. The equation becomes: \[ 2 \log(8n+4) = 4 \]

Step 2: Divide by 2

Divide both sides of the equation by 2 to further isolate the logarithm:\[ \log(8n+4) = 2 \]

Step 3: Convert Logarithmic Equation to Exponential Form

Convert the logarithmic equation to its exponential form. Recall that \( \log_b(x) = y \) implies \( b^y = x \). Here, we assume the base to be 10 since no base is specified:\[ 10^2 = 8n+4 \]

Step 4: Solve for n

Now, solve the exponential equation for \( n \).\[ 100 = 8n + 4 \]Subtract 4 from both sides:\[ 96 = 8n \]Divide both sides by 8:\[ n = 12 \]

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics that help us solve for unknowns in exponential equations. A logarithm answers the question, "To what number must we raise the base to get a certain value?" For instance, in the equation \( \log_b(x) = y \), \( b \) is the base, \( x \) is the value you're trying to find the log of, and \( y \) is the exponent or the result of the logarithm.

It's important to remember:

• Common bases include 10 (common logarithm) and \( e \) (natural logarithm).
• Notations like \( \log(x) \) typically mean \( \log_{10}(x) \) unless another base is specified.
• Logarithms can be used to bring down exponents, which makes them useful for solving exponential equations.

To solve logarithmic equations, we often manipulate the equation to isolate the logarithm, and then convert to exponential form. This helps in simplifying and eventually solving the equation.

Exponential Form

Exponential form allows us to express logarithmic equations as exponential ones, which can be easier to solve. The conversion is based on the fundamental relationship: \( \log_b(x) = y \) implies \( b^y = x \). In simpler terms:

• \( b \) is the base of the exponential expression.
• \( y \) is the power or exponent we raise the base to.
• \( x \) is the result or value obtained.

In the given problem, we converted \( \log(8n+4) = 2 \) into its exponential form \( 10^2 = 8n+4 \).

This step is crucial because it allows us to work directly with the exponential expression, which is generally straightforward to solve by standard algebraic methods, such as simplifying and isolating the variable.

Solving Equations

When solving equations that involve logarithms, the goal is to find the variable's value that makes the equation true. We use a systematic approach:

• **Isolate the logarithmic part:** Just like any variable or term in an equation, start by getting the logarithmic term on one side. This usually involves basic algebra, such as subtracting or dividing.
• **Convert to exponential form:** Once isolated, rewrite the logarithmic equation as an exponential one. This is especially helpful when the base is 10, as calculations become simpler.
• **Solve the new equation:** With the equation in exponential form, use algebraic techniques to solve for the variable. This often involves simplifying expressions and performing basic operations like addition, subtraction, multiplication, or division.

This approach, highlighted in the example equation \( 2 \log (8n+4) + 6 = 10 \), shows each essential step—from isolating the logarithmic term to solving for \( n = 12 \).

Each step must be followed precisely to ensure a correct solution, demonstrating the importance of understanding the relationships between logarithmic and exponential forms.