Problem 42 - Algebra and Trigonometry

Question

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{2 x+1}=3^{x+2} $$

Step-by-Step Solution

Answer

The exact solution for x is \(\frac{2\ln(3) - \ln(7)}{(2 \ln(7) - \ln(3))}\). Use a calculator to find an approximate decimal value, rounded to two decimal places.

Step 1: Write the equation

Start with the original equation, \(7^{2 x+1}=3^{x+2}.\)

Step 2: Apply natural logarithm to both sides

Applying natural logarithm (ln) to both sides will assist in simplifying the equation thus, \(\ln(7^{2x+1})=\ln(3^{x+2}).\)

Step 3: Use properties of logs to bring down exponents

The properties of logarithms state that \(\ln(a^b)=b\ln(a)\). Using this, the equation can be rewritten as \((2x+1)\ln(7) = (x+2)\ln(3)\).

Step 4: Solve for x

First expand the equation to \(2x \ln(7) + \ln(7) = x \ln(3) + 2 \ln(3)\). Then collect all terms of x on one side and the constants on the other, \((2x \ln(7) - x\ln(3)) = (2\ln(3)-\ln(7))\). Divide both sides by the coefficient of x to isolate x. Thus, \(x = \frac{2\ln(3) - \ln(7)}{(2 \ln(7) - \ln(3))}\). This is the exact solution of x. Use a calculator to find the decimal approximation.

Step 5: Decimal approximation of x

Plug in the natural logarithms values in a calculator to obtain a decimal approximation for x. Round off the value to two decimal places.

Key Concepts

Natural Logarithms

In the context of solving exponential equations, understanding natural logarithms is crucial. A natural logarithm is the logarithm to the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. The notation used for natural logarithms is 'ln'. For example, when you see the expression \(\ln(x)\), it's asking for the power to which 'e' must be raised to get the number 'x'.

Natural logarithms are particularly useful for solving equations where the variable is in an exponent, which is why they appear in the process of solving the given problem. The intrinsic properties of natural logarithms are applied to simplify the equations and make it possible to isolate the variable, allowing one to solve for its value.

Properties of Logarithms

Logarithms have special properties that make them invaluable for solving exponential equations. One such property is the 'Power Rule', stating that \(\ln(a^b) = b \ln(a)\), which allows the exponent in a logarithmic function to be brought in front as a multiplier. This rule is applied to transform the given exponential equation into a form where the variable 'x' can be isolated more easily.

Other properties include the 'Product Rule' \(\ln(ab) = \ln(a) + \ln(b)\), the 'Quotient Rule' \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\), and the 'Change of Base Formula', which may also be used when solving logarithmic equations. Familiarity with these rules is necessary to effectively manipulate and solve equations involving logarithms.

Decimal Approximation

After finding the exact solution of an equation involving natural logarithms, the next step is often to compute a decimal approximation for more practical applications. A decimal approximation is a rounded value that represents an irrational number in a finite and practical way. Calculators or mathematical software can be used to compute these approximations to the desired degree of accuracy.

In the case of our problem, rounding off to two decimal places is requested. This level of precision is often enough for many real-life situations and is standard practice for homework assignments or tests, providing a balance between accuracy and usability.

Exponential Equation Steps

Solving exponential equations involves a series of logical steps that harnesses the power of logarithms. Here's how it unfolds:

• Isolate the exponential expression: Start by ensuring the equation is set up so the exponential expressions are on opposite sides of the equation.
• Apply logarithms: Use natural logarithms to both sides to simplify the exponents' handling.
• Use log properties: Apply properties of logarithms, such as the Power Rule, to bring down exponents and make the variable accessible.
• Solve for the variable: Isolate the variable using algebraic manipulations. You may need to collect like terms or factor as necessary.
• Find a decimal approximation: Finally, use a calculator to approximate the solution as a decimal, rounding to the appropriate degree of precision.

By following these steps and understanding their underlying reasoning, exponential equations become much more manageable.