Problem 39 - 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. $$ 7^{0.3 x}=813 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 36.83\)

Step 1: Rewrite the Equation

Write the given equation \(7^{0.3 x}=813\) in its raw form.

Step 2: Applying Logarithms

Apply natural logarithm (ln) to both sides of the equation. This gives us \(\ln (7^{0.3 x})=\ln(813)\). The property of logs allows us to move the exponent of the log into the front which simplifies the equation to \(0.3 x*\ln(7)=\ln (813)\).

Step 3: Solve for x

Now isolate x by dividing each side of the equation by \(0.3 * \ln(7)\), i.e, \(x=\ln(813)/ (0.3 * \ln(7))\).

Step 4: Decimal Approximation

Use a calculator to compute the decimal approximation of x up to two decimal places. This results in \(x \approx 36.83\)

Key Concepts

Understanding Natural Logarithms

Natural logarithms are a type of logarithm that uses the base of the mathematical constant e (approximately 2.71828). They are denoted by the "ln" symbol. Natural logs are particularly useful because they simplify complex exponential functions, making them easier to solve.

When you see an equation like \(7^{0.3 x} = 813\), applying the natural logarithm allows you to bring down the exponent as a multiplier. This is possible due to the logarithmic identity that states \(\ln(a^b) = b \cdot \ln(a)\). By applying the natural log to both sides of the equation, we simplify it to \(0.3 x \cdot \ln(7) = \ln(813)\).

• Natural logarithms help convert multiplicative relationships into additive ones.
• This property is crucial for solving exponential equations.

In this way, natural logarithms are a powerful tool for solving equations where the variable is an exponent.

Exploring Common Logarithms

Common logarithms use the base 10 and are expressed with the "log" symbol, without a subscript. While not used in our current problem, they're often used in scientific and real-world applications where powers of ten are common.

They are advantageous because they help simplify calculations and make estimations clearer, especially when dealing with very large or small numbers. Like natural logs, they can transform exponential equations into linear ones by leveraging the property \( \log(a^b) = b \cdot \log(a) \).

• They provide an accessible way to solve equations involving powers and roots of ten.
• Useful in contexts like pH calculations, sound intensity, and certain financial calculations.

Even though natural logarithms are more widely used for expressing exponential growth, common logarithms are still a fundamental mathematical tool.

Using Decimal Approximation

Decimal approximation is the process of finding a number close to an exact solution but expressed as a decimal. Since exact values of exponential equations can be very complex and unwieldy, especially in scientific fields or practical applications, a decimal approximation is often considered sufficient.

Once the solution to an equation like \(x = \frac{\ln(813)}{0.3 \cdot \ln(7)}\) is found, it's important to use a calculator to get a value that can be easily interpreted. In this instance, solving this expression, we find \(x \approx 36.83\), rounded to two decimal places.

• Decimal approximations provide quick and practical solutions to elaborate calculations.
• They allow us to comprehend and communicate our findings more effectively.

Approximations facilitate understanding and ensure that results are communicated clearly and promptly, especially in complex equations.