Problem 28 - Algebra 2

Question

Use natural logarithms to solve each equation. $$ e^{\frac{x}{9}}-8=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 9 * \ln 14\)

Step 1: Start by isolating the exponential term

To do this, add 8 to each side of the equation to get: \(e^{\frac{x}{9}}=14\)

Step 2: Convert the equation to logarithmic form

A property of logarithms allows us to write this exponential equation as a logarithmic equation. So, the equation now becomes: \(\frac{x}{9}= \ln 14\)

Step 3: Solve for x

To get \(x\) alone, multiply each side of the equation by 9. This gives the final solution: \(x = 9 * \ln 14\)

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. In our exercise, the equation is in the form of \( e^{\frac{x}{9}} - 8 = 6 \), where the base is the mathematical constant \( e \) (approximately 2.718). This type of equation is significant because exponential equations often represent real-world phenomena such as population growth, radioactive decay, and compound interest.

In this exercise, the key to solving the equation is **isolating the exponential part**. First, we added 8 to both sides to remove the constant from the left side, simplifying the equation to \( e^{\frac{x}{9}} = 14 \). Keeping the base \( e \) on one side is crucial for the next step, where we convert the exponential form to a logarithmic form.

Logarithmic Equations

Logarithmic equations may not look as familiar as exponential ones, but they are equally important. These equations involve logarithms, which are the inverse operations of exponentiation. In essence, logarithms ask "what power must a base be raised to get a given number?"

For our exercise, we needed to transform the exponential equation \( e^{\frac{x}{9}} = 14 \) into its logarithmic form. By using the natural logarithm, we translate this into \( \ln 14 = \frac{x}{9} \). The natural logarithm is particularly useful with exponential functions, especially those involving \( e \), due to their shared properties. The conversion to a logarithmic equation helps simplify the solution process by making the exponent the subject of the equation.

Solving Equations Using Logarithms

Solving equations using logarithms is practical and efficient when dealing with exponential terms. Once we have the logarithmic form \( \ln 14 = \frac{x}{9} \), our objective is to isolate \( x \).

We accomplish this by multiplying both sides by 9, removing the fraction and isolating \( x \). This approaches solving like we would solve simpler linear equations, where adjusting multipliers or divisors helps isolate the variable. In this case, multiplying by 9 gives us the answer. Therefore, \( x = 9 \times \ln 14 \). This expression provides the numeric value for \( x \) once evaluated using a calculator to find \( \ln 14 \).

Using logarithms simplifies the often complex task of solving exponential equations by transforming them into a more manipulatable form, making solutions more accessible.

Properties of Logarithms

Understanding the properties of logarithms is key to successfully using them to solve equations. These properties make logarithmic equations versatile tools in mathematics:

• **Product Rule:** \( \ln(ab) = \ln a + \ln b \).
• **Quotient Rule:** \( \ln(\frac{a}{b}) = \ln a - \ln b \).
• **Power Rule:** \( \ln(a^b) = b \cdot \ln a \).

In our example, these properties come in handy because they logically approach how exponentiation would normally work, but in a different mathematical form.

Though we didn't directly apply all these properties to solve our specific exercise, they underpin the transformation process from an exponential to a logarithmic equation. For instance, having knowledge of the power rule can help us understand why \( \ln 14 = \frac{x}{9} \) effectively captures the essence of the original exponential equation, making the solution possible.