Problem 18 - College Algebra

Question

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$3^{x / 7}=0.2$$

Step-by-Step Solution

Answer

The solution in terms of natural logarithms is \(x = 7 \times (\ln 0.2 / \ln 3)\), and the decimal approximation, if calculated using a calculator, would provide the final numeric answer.

Step 1: Express as a logarithm

The exponential equation \(3^{x / 7}=0.2\) can be written as a logarithm. This can be done using the logarithmic form \(b^a=c\) is equivalent to \(\log_b c = a\). So, the equation becomes \(\log_3 0.2 = x/7\).

Step 2: Converting to natural logarithm

The natural logarithm form of this expression can be found by using the change of base formula. This formula, defined as \(\log_b a = \ln a / \ln b\), allows us to convert \(\log_3 0.2\) into \(\ln 0.2 / \ln 3 = x / 7\).

Step 3: Solve for x

To solve for x, multiply both sides by 7, to get \(x = 7 \times (\ln 0.2 / \ln 3)\).

Step 4: Decimal Approximation

Finally, for obtaining the decimal approximation, use a calculator to calculate the value of \(7 \times (\ln 0.2 / \ln 3)\) giving the result correct to two decimal places.

Key Concepts

Natural Logarithms

When dealing with exponential equations, a natural logarithm, denoted as \( \ln \), is an invaluable tool. Unlike common logarithms that are based on the number 10, natural logarithms use the irrational number 'e' (approximately 2.71828) as their base. The equation \( y = \ln x \) can be interpreted as 'e raised to the power of y equals x'. Among its various properties, one significant feature is that the natural logarithm of 1 equals zero, because any number to the power of zero equals one, and this is also true for the base 'e'.

Use of the natural logarithm is especially helpful when you face an unknown exponent that can't easily be simplified using base 10. Given the relationship between 'e' and natural logarithms, when an exponential equation has a base other than 10 or 'e', we express the solution using natural logarithms because they simplify the calculations involved in finding the exponent.

Change of Base Formula

The change of base formula is a lifeline for solving exponential equations with bases other than 10 or 'e'. It states that to convert an expression with a base 'b' to a new base 'k', you use the formula \( \log_b a = \frac{\log_k a}{\log_k b} \). An important application of this is transforming log equations into a form that can be handled by calculators, which typically have keys for base 10 (log) and base 'e' (ln) logarithms.

In the context of our original example, \( 3^{x/7} = 0.2 \), we can't directly find \( x \) because most calculators can't compute \( \log_3 0.2 \) directly. By invoking the change of base formula, we seamlessly convert it into an expression with natural logarithms: \( \log_3 0.2 = \frac{\ln 0.2}{\ln 3} \). This nifty transformation allows for a straightforward calculation of \( x \) with a tool that's accessible to most students—a scientific calculator.

Decimal Approximation

After using natural logarithms and the change of base formula to express an exponential equation, the final step is often to approximate the answer to a decimal value. Calculators are generally used for this step, providing a practical numeric approximation to a given number of decimal places.

For instance, in our problem, after applying the change of base formula and isolating \( x \), the exact expression we have is \( x = 7 \times (\ln 0.2 / \ln 3) \). However, this form isn't very 'friendly' for understanding at a glance or using in further calculations. By using a calculator to find the decimal approximation of the natural logarithm values and performing the multiplication, the answer can be presented in a simple, straightforward decimal form, such as \( x = -13.54 \) when rounded to two decimal places. This process illustrates how we can bridge the gap between abstract mathematical concepts and practical, real-world applications.