Problem 36 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. \(8+3 e^{3 x}=26\)

Step-by-Step Solution

Answer

\( x \approx 0.5973 \)

Step 1: Move Constants

First, isolate the exponential term on one side of the equation by subtracting 8 from both sides of the equation: \[ 3e^{3x} = 26 - 8 \] Simplifying this gives:\[ 3e^{3x} = 18 \]

Step 2: Divide by Coefficient

Next, divide both sides by 3 to solve for \( e^{3x} \): \[ e^{3x} = \frac{18}{3} \] Simplifying the right side gives:\[ e^{3x} = 6 \]

Step 3: Take the Natural Logarithm

To remove the exponential, take the natural logarithm (ln) on both sides of the equation:\[ \ln(e^{3x}) = \ln(6) \] Applying the property \( \ln(e^y) = y \), we have:\[ 3x = \ln(6) \]

Step 4: Solve for x

Divide both sides by 3 to solve for \( x \): \[ x = \frac{\ln(6)}{3} \] Using a calculator, compute \( \ln(6) \) and perform the division. Rounding to the nearest ten-thousandth gives:\[ x \approx 0.5973 \]

Key Concepts

Understanding Natural Logarithms

The natural logarithm, denoted as \( \ln \), is a special type of logarithm with a base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is widely used in mathematics, especially when dealing with exponential functions. It helps in solving equations involving the exponential function \( e^x \) by applying the inverse operation.

A key property of the natural logarithm is that \( \ln(e^y) = y \). This property is helpful because it allows us to transform an exponential equation into a linear one, simplifying the solving process significantly. When you take the natural logarithm of both sides of an equation, it is crucial to maintain the equality and apply it correctly to each component.

• The expression \( \ln(e^{3x}) = \ln(6) \) uses this principle to extract the exponent 3x.
• It's important to note that \( \ln(e) = 1 \), which simplifies calculations when \( e \) is raised to a power.

In exponential equations, using the natural logarithm effectively is key to unravel complex relationships swiftly.Additionally, knowing how to apply logarithmic properties, such as the product, quotient, and power rules, is essential for handling more complicated equations.

Rounding Decimals Accurately

Rounding decimals is an important skill in mathematics, particularly in contexts where precise solutions are needed, such as in solving equations. When you round to the nearest ten-thousandth, it means you're looking at the fourth decimal place.

Here’s a quick guide:

• Identify the digit in the ten-thousandth place (the fourth digit after the decimal point).
• Look at the next digit (the fifth digit). If this digit is 5 or greater, increase the ten-thousandth digit by 1.
• If it is less than 5, retain the ten-thousandth digit as it is.

For the solution \( x \approx 0.5973 \), we look at the third decimal, 7, and behind it is a number less than 5, thus no rounding action is necessary. Hence the solution is appropriately \( x \approx 0.5973 \).

Rounding accurately ensures consistency in problem-solving processes and is particularly vital when the precision of decimal points affects an equation's viability or interpretation. It's an essential skill to refine and maintain consistency across your calculations.

Solving Exponential Equations

Exponential equations are equations where a variable appears in the exponent. These are particularly interesting because they can grow very rapidly with small changes in the exponent.

The first step in solving such equations, as seen in our original problem, is isolating the exponential expression. This often involves basic algebraic steps like moving terms across the equation or dividing both sides by a coefficient.

Once the exponential term is isolated, the next key step is to "undo" the exponential function. For equations involving \( e^x \), this is usually done by taking the natural logarithm, since the natural logarithm's base is \( e \) and can effectively reverse the exponential function.

• Transformation: Converting \( e^{3x} = 6 \) using \( \ln(e^{3x}) = \ln(6) \) results in the simplification \( 3x = \ln(6) \).
• Isolating the variable: You'll then solve for \( x \) by isolating it, usually by further algebraic manipulation like division, giving \( x = \frac{\ln(6)}{3} \).

Solving exponential equations involves understanding and using logarithmic properties to make the problem manageable and finding precise solutions. This process is foundational for tackling more complex mathematical and real-world problems that involve exponential growth or decay. Mastery of these steps provides a strong groundwork for related advanced topics.