Problem 28 - College Algebra

Question

Solve each exponential equation in Exercises \(23-48\). 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. $$ 19^{x}-143 $$

Step-by-Step Solution

Answer

The logarithmic solution for \(x\) is \(\dfrac{ln(143)}{ln(19)}\), and the decimal approximation will be calculated using a scientific calculator, but for programming you might simply use (import math: math.log(143) / math.log(19)).

Step 1: Isolate the exponential term

Firstly, the exponential term, which is \(19^{x}\), needs to be isolated on one side of the equation. This equation \(19^{x}-143 = 0\) can be rewritten as \(19^{x} = 143\).

Step 2: Take the natural logarithm of both sides

Next, use the natural logarithm, labeled as ln, on both sides of the equation to simplify the equation to a state where \(x\) can be solved. Remembering that \(ln(a^{b}) = b \cdot ln(a)\), the equation becomes \(ln(19^{x}) = ln(143)\), which simplifies to \(x \cdot ln(19) = ln(143)\).

Step 3: Solve for x

To solve the equation for \(x\), divide both sides of the equation by \(ln(19)\) to isolate \(x\), meaning \(x = \dfrac{ln(143)}{ln(19)}\).

Step 4: Calculate decimal approximation

Finally, to find a decimal approximation for \(x\), use a calculator to divide \(ln(143)\) by \(ln(19)\). Make sure to round to two decimal places.

Key Concepts

Natural Logarithms

Natural logarithms, often denoted as \( \ln \), are a crucial mathematical tool used to unravel complex exponential equations. The natural logarithm is the logarithm to the base \( e \), where \( e \) is approximately 2.718. It's called "natural" because it frequently appears in natural growth processes. In the exercise, we take the natural logarithm of both sides of the equation \( 19^{x} = 143 \) to simplify our work with exponential expressions. When you apply a natural logarithm to both sides, the power rule \( \ln(a^{b}) = b \cdot \ln(a) \) comes into play. This rule allows us to bring down the exponent, \( x \), making it linear and easier to solve. By transforming the equation into \( x \cdot \ln(19) = \ln(143) \), we've moved a step closer to solving for \( x \). Always remember, the primary goal of using natural logarithms is to linearize an exponential equation by turning the exponent into a multiplicative factor.

Solving Equations

Solving equations entails finding the value of variables that satisfy the equation. The method for solving an exponential equation begins with isolating the exponential part. For the equation \( 19^{x} = 143 \), we first ensure \( 19^{x} \) is on one side. By taking the natural logarithm of both sides, we exploited the power rule to extract \( x \) from the exponent, resulting in \( x \cdot \ln(19) = \ln(143) \). To solve for \( x \), our goal is to isolate it. We do this by dividing each side by \( \ln(19) \). Hence, \( x = \frac{\ln(143)}{\ln(19)} \). Solving in this manner highlights a key principle: breaking down exponents into more manageable linear equations allows us to systematically solve for unknowns.

Decimal Approximation

Decimal approximation involves finding a numerical estimation close to the exact value of a mathematically derived solution. In our exercise, after determining the solution \( x = \frac{\ln(143)}{\ln(19)} \), we transition to decimals for practicality. To find the decimal approximation, we use a calculator to evaluate \( \ln(143) \) and \( \ln(19) \), then perform the division. It's essential to follow the instruction of rounding the resultant quotient to two decimal places, ensuring precision is met while maintaining a balance with readability. For many real-world applications, such approximations are not just convenient; they are necessary, as they provide solutions in a format amenable to further computational steps or interpretive contexts. Always verify your approximation for any critical errors by considering whether it reasonably fits within the expected range of the calculated solution.