Problem 579 - Precalculus

Question

Use the definition of a logarithm to find the exact solution for \(4 \log (2 n)-7=-11\)

Step-by-Step Solution

Answer

The solution is \( n = 0.05 \).

Step 1: Isolate the Logarithm

First, let's isolate the logarithmic expression in the equation. Starting with:\[ 4 \log(2n) - 7 = -11 \]Add 7 to both sides:\[ 4 \log(2n) = -4 \]

Step 2: Divide to Simplify

Next, divide both sides of the equation by 4 to isolate the logarithm:\[ \log(2n) = -1 \]

Step 3: Convert Logarithmic to Exponential Form

To solve for \( n \), remember that \( \log(b) = x \) implies \( 10^x = b \). Here, \( \log(2n) = -1 \) implies:\[ 10^{-1} = 2n \]

Step 4: Solve for n

Simplify \( 10^{-1} \) to get \( 0.1 \). Then solve for \( n \) by dividing both sides by 2:\[ n = \frac{0.1}{2} \]\[ n = 0.05 \]

Key Concepts

Isolation of Logarithmic Expressions

When solving equations that include logarithms, like in the example equation \(4 \log(2n) - 7 = -11\), a vital first step is to isolate the logarithmic expression. This process helps to simplify the equation and makes it easier to handle. To do this, we need to move all non-logarithmic terms to the other side of the equation.

In our example, we start by adding 7 to both sides to clear the constant:

• \(4 \log(2n) - 7 + 7 = -11 + 7\)
• Leaving us with \(4 \log(2n) = -4\).

After this step, we focus on getting the logarithmic part \(\log(2n)\) by itself. We do this by dividing each term by 4

• \(4 \log(2n) / 4 = -4 / 4\)
• Resulting in \(\log(2n) = -1\).

Once the logarithmic expression is isolated, we can proceed to the next step.

Converting Logarithms to Exponents

With the logarithmic term isolated as \(\log(2n) = -1\), the next step involves converting this logarithmic expression into its equivalent exponential form. This step is crucial because it allows for the straightforward solution of the equation.

The relationship between a logarithm and its exponential form is defined as \(\log_b(y)=x\) meaning \(b^x = y\). In our case, it means:

• Given \(\log(2n) = -1\), translate to exponential form:
• \(10^{-1} = 2n\)

Converting it this way simplifies understanding based on powers of ten. The exponential form lays out that \(2n\) is calculated by raising 10 to the power of \(-1\), which is straightforward to solve in a subsequent step.

Solving Equations

Once the equation \(10^{-1} = 2n\) is in an exponential form, solving it becomes straightforward. This step is where the values for variables are calculated directly.

Start by evaluating the power of ten:

• \(10^{-1} = 0.1\)

With this simplified to \(0.1 = 2n\), divide both sides by 2 to isolate \(n\):

• \(n = \frac{0.1}{2}\)
• \(n = 0.05\)

And there you have it; \(n\) is found to be \(0.05\). This fine-tuned method seamlessly flows from isolating the logarithm to calculating the final value.