Problem 29 - 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. $$ 5 e^{x}=23 $$

Step-by-Step Solution

Answer

After going through these steps, the approximation we arrive at is \(x\approx 0.80\).

Step 1: Isolate the exponential term

To isolate the exponential term, divide both sides of the equation by 5. The equation becomes \(e^{x} = \frac{23}{5}\).

Step 2: Apply logarithm

Apply the natural logarithm to both sides of the equation. The natural logarithm of \(e^{x}\) is just \(x\) (since \(e\) is the base of the natural logarithm), and so the equation becomes \(x = \ln(\frac{23}{5})\).

Step 3: Simplify using a calculator

Make use of a calculator to find the decimal approximation of \(\ln(\frac{23}{5})\). Make sure to round to two decimal places.

Key Concepts

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the number \( e \), which is approximately 2.71828. This mathematical constant \( e \) is irrational, meaning it cannot be expressed as a simple fraction and has an infinite number of decimal places without repeating.

When solving exponential equations involving \( e \), such as the equation \( e^x = \frac{23}{5} \), natural logarithms are particularly useful. By applying the natural logarithm (denoted as \( \ln \)) to both sides of the equation, we utilize the property that \( \ln(e^x) = x \). This simplifies our problem and makes it easier to extract \( x \) from the exponential term.

The natural logarithm is a powerful tool in calculus and many areas of mathematics due to its inherent relationship with exponential functions and its mathematical properties.

• The natural logarithm of \( e \) is 1, i.e., \( \ln(e) = 1 \).
• \( \ln(1) = 0 \) because any number raised to the power of 0 is 1.
• It allows for the straightforward manipulation of equations involving growth and decay, like population growth or radioactive decay.

Common Logarithms

Common logarithms are logarithms that have a base of 10. They are often denoted as \( \log \) without a base indicated, assuming the base is 10. Common logarithms are widely used in scientific notation and are the default when using logarithmic tables or certain calculators.

While natural logarithms are often used for equations involving \( e \), common logarithms provide a useful method when dealing with base 10 calculations, often simplifying steps in various algebraic and scientific computations.

• \( \log(10) = 1 \), because 10 raised to the power of 1 is 10.
• \( \log(1) = 0 \), because any number raised to the power of 0 is 1.
• Common logarithms are particularly useful in the fields of engineering, physics, and acoustics.

To transform a solution from a natural logarithm into a common logarithm context, use the formula \( \ln(x) = \frac{\log(x)}{\log(e)} \), where \( \log(e) \) can be determined using a calculator.

Decimal Approximation

Decimal approximation is the process of approximating the value of a number using decimal forms, which is especially helpful when an exact form is impractical or overly complex. In the context of solving exponential equations, once you have expressed the solution in logarithmic form, a calculator can help find a numerical value.

For example, after solving an equation and finding \( x = \ln(\frac{23}{5}) \), you would use a calculator to find this value in decimal form. For accuracy and consistency, it is important to round the result to the desired number of decimal places—two in this specific case.

Here's how to perform a decimal approximation:

• Calculate \( \ln(\frac{23}{5}) \) using a calculator.
• Observe the decimal output.
• Round off to two decimal places to maintain precision as requested.

Decimal approximations are beneficial for quick estimations and are widely accepted in practical applications and when making real-world calculations.