Problem 17 - Algebra 2

Question

Solve each equation. Check your answers. $$ \ln (2 m+3)=8 $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln (2m + 3) = 8 \) is \( m \approx 1488.98 \).

Step 1: Convert the Logarithmic Equation to an Exponential Equation

To convert \(\ln (2m + 3) = 8\) to an exponential equation, remember the base of natural logarithm is \( e \). Thus, \( e^8 = 2m + 3 \).

Step 2: Solve the Exponential Equation for \( m \)

First, simplify \( e^8 \) to get its approximate value, which is 2980.96. Therefore, the equation becomes 2980.96 = 2m + 3. To solve for \( m \), subtract 3 from both sides of the equation to get 2977.96 = 2m. Then, divide both sides by 2 to find \( m \), which is 1488.98.

Step 3: Check the solution

To verify the solution, substitute it back into the original logarithmic equation \(\ln (2m + 3) \). If the left-hand side is equal to the right-hand side (8), then the solution is correct. Substituting \( m = 1488.98 \) into the equation gives approximately \( \ln (2978) \), and the natural logarithm of 2978 is approximately 8, which verifies the solution.

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithms that use the constant \( e \) as their base.\( e \) is an irrational number approximately equal to 2.71828. The notation for a natural logarithm is \( \ln \), so \( \ln(x) \) denotes the natural logarithm of \( x \). This is used frequently in calculus and higher mathematics because of its convenient mathematical properties.Natural logarithms are useful in solving equations involving exponential growth and decay. They are also helpful in transforming equations to make them easier to handle. For instance, if you encounter \( \ln (2m + 3) = 8 \), the natural logarithm makes it possible to rewrite this in exponential form as \( e^8 = 2m + 3 \). This transformation can make the equation much simpler to work with.

• Why Use \( \ln \)? The natural logarithm simplifies dealing with expressions where variables are in exponents.
• What Is \( e \)? It is the base of the natural logarithm, and it is a fundamental constant in mathematics, particularly in calculus.

Understanding how to use natural logarithms is an essential skill in solving logarithmic equations, as it helps translate complex relationships into comprehensible formats.

Exponential Equations

Exponential equations are equations in which variables appear as exponents. A typical exponential equation will look like \( a^x = b \), where \( a \) and \( b \) are known numbers, and you solve for the unknown \( x \). When solving the exponential equivalents of logarithmic equations, it's essential to know how to manipulate them properly.In the given exercise, converting \( \ln (2m + 3) = 8 \) to an exponential equation results in \( e^8 = 2m + 3 \). This bridge from logarithmic to exponential form is crucial: it leverages the power of exponents to allow us to isolate and solve for variables.

• Importance of Exponential Equations: They appear in many areas, including scientific contexts involving growth and decay models, like population growth or radioactive decay.
• Solving Techniques: After transforming logarithmic forms into exponential equations, the simplification process often involves basic algebraic manipulations like subtraction and division.
• Tips: Always remember to check the feasibility of your answer. After solving for \( x \) (or any variable), substitute back into the original equation to confirm correctness.

The practice of using exponential equations in conjunction with logarithmic ones enables a deeper understanding of growth processes and mathematical modeling.

Checking Solutions

After solving any logarithmic or exponential equation, it's essential to verify your solution. This ensures that your solution is valid within the context of the original problem. In the case of our given equation \( \ln (2m + 3) = 8 \), checking the solution involved substituting \( m = 1488.98 \) back into the equation to ensure it holds true.The process of checking solutions involves evaluating the original expression with your found value and comparing it to the expected result. If both sides of the equation match, your solution is verified.

• Why Check? Checking solutions identifies errors and confirms correctness, especially when dealing with complex transformations or approximations.
• How to Check Effectively: Substitute the solution back into the original form of the equation. Calculate the left-hand and right-hand sides independently to ensure they are equal.
• Common Mistakes: Pay attention to precise calculation and rounding errors when dealing with approximations.

Ensuring that your solution satisfies the original equation gives confidence in your mathematical reasoning and outcomes.