Problem 40 - College Algebra Essentials

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 short answer for x, correct to two decimal places, would be obtained from the calculation in Step 4. The result should be written here after performing the calculations.

Step 1: Transform the exponential equation into a logarithmic equation

By transforming, the initial equation \[3^{\frac{x}{7}}=0.2\] can be rewritten as \[\log_{3}0.2=\frac{x}{7}\] using logarithmic properties.

Step 2: Isolate the variable

Isolate x by multiplying both sides by 7, therefore \[x=7\log_{3}0.2\].

Step 3: Calculate decimal approximation

Now, use a calculator to obtain a decimal approximation. It should be mentioned that the natural logarithm might be more available on standard calculators. Conversion can be made using the formula \[ \log_{a}b=\frac{\ln{b}}{\ln{a}} \]}, and this gives \[ x=7\frac{\ln{0.2}}{\ln{3}} \]

Step 4: Obtain the final decimal approximation

Using the calculator gives the final decimal approximation, correct to two decimal places.