Problem 40 - 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. $$ 3^{\frac{x}{7}}=0.2 $$

Step-by-Step Solution

Answer

The solution for the given exponential equation is \(x \approx -7.11\).

Step 1: Rewrite in logarithmic form

The equation \(3^{\frac{x}{7}} = 0.2\) can be rewritten in logarithmic form as\(\log_3(0.2) = \frac{x}{7}\). This uses the property of logarithms that \(b^y = x\) is equivalent to \(\log_b(x) = y\). This is a key step to make the problem easier to solve for x.

Step 2: Solve for x

Multiply both sides by 7 to solve for x: \(x = 7 \times \log_3(0.2)\). This uses the principle of equality - what you do to one side of the equation, you must do to the other.

Step 3: Compute the decimal approximation

A calculator can be used to find the logarithmic and multiplication result. The answer should be correct to two decimal places: \(x \approx -7.11\). The negative result is because 0.2 is less than 1, and any number (other than 1) to a negative power is less than 1.

Key Concepts

Logarithmic Form

To solve exponential equations like \(3^{\frac{x}{7}} = 0.2\), we often convert them into a logarithmic form. Logarithmic form helps in isolating the variable which is an exponent in the equation.
An important property of logarithms is that the equation \(b^y = x\) is equivalent to \(\log_b(x) = y\). This property allows us to rewrite our equation as \(\log_3(0.2) = \frac{x}{7}\). This conversion simplifies the equation significantly, making it easier to solve for \(x\).

• In the initial equation, the base \(3\) and \(0.2\) are arguments for the logarithm, and \(\frac{x}{7}\) becomes the result of the logarithm.
• This step transforms the perhaps daunting power problem into a more straightforward algebraic task.

Understanding this conversion process is fundamental to solving logarithmic and exponential equations efficiently. Always remember the interchangeability between exponential and logarithmic forms.

Natural Logarithms

Natural logarithms use the constant \(e\), approximately equal to 2.718, as their base. However, in solving equations like \(3^{\frac{x}{7}} = 0.2\), it's more common to use natural logarithms as tools in calculation through the natural log \(\ln\) function on calculators.
Use the change of base formula if needed to convert between different logarithmic forms, \[\log_3(0.2) = \frac{\ln(0.2)}{\ln(3)}\]This step can sometimes be required in calculations where calculators only accept natural or common logarithms. Always verify whether switching to logarithms with base \(e\) can speed up the calculation.

• Natural logs are prevalent in many fields like calculus, since the constant \(e\) naturally arises in various applications including compound interest and change rates.
• Ensure you're comfortable with shifting between logarithmic bases if the situation demands it.

Mastering natural logarithms and their application in problem-solving will greatly enhance your mathematical toolkit.

Decimal Approximation

After solving the equation \(x = 7 \times \log_3(0.2)\), we arrive at the next step—finding a decimal approximation. Calculators are typically used here to compute a more exact numeric representation done to a specific precision, often two decimal places.
Our exact numerical answer becomes \(x \approx -7.11\).
The process involves:

• Calculating the logarithm \(\log_3(0.2)\) with the change of base formula if necessary.
• Multiplying the result by 7 to get the value of \(x\).
• Round to the needed decimal accuracy based on the problem's requirements. Often, the answer is rounded to two decimal places.

It's essential to understand why the value of \(x\) is negative. Since \(0.2 < 1\), a positive base raised to a negative power results in a number less than one. That reasoning explains why our calculated \(x\) is negative. Recognizing rounding conventions and maintaining accuracy in your calculations is crucial in achieving the right solution.