Problem 11 - 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. $$e^{1-5 x}=793$$

Step-by-Step Solution

Answer

The solution for 'x' in terms of natural logarithms is \(x = -(\ln(793) - 1) / 5\). The decimal approximation of 'x', correct to two decimal places, needs to be calculated using a calculator.

Step 1: Write down the equation

The equation to solve is \(e^{1-5x} = 793\).

Step 2: Apply natural logarithm to both sides

The exponential equation can be transformed to logarithmic form using the rule \(log_b a = c \iff b^c = a\). Apply the natural logarithm (ln) to both sides, using the fact that ln(e) = 1, to obtain the equation \ln(e^{1-5x}) = \ln(793)\, or equivalently, \((1-5x)\ln(e) = \ln(793)\). This simplifies to \(1 - 5x = \ln(793)\).

Step 3: Solve for x

Rearrange the equation to solve for 'x' by first subtracting '1' from both sides, making the equation \(-5x = \ln(793) - 1\). Then, divide by '-5' to find \(x = -(\ln(793) - 1) / 5\).

Step 4: Calculate decimal approximation

Use a calculator to find the numerical value of \(x = -(\ln(793) - 1) / 5\) correct to two decimal places.

Key Concepts

Natural Logarithms

When we talk about natural logarithms, we are referring to the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. This particular base of logarithm is denoted as 'ln' and plays a crucial role in mathematics due to its natural appearance in calculus and mathematical modeling.

A key property of the natural logarithm function is that it is the inverse of the exponential function with base e. So, if we have an exponential equation like e raised to some power equals a number—symbolically, e^y = x—the natural logarithm allows us to solve for this power, which can be represented as ln(x) = y. This process is indispensable when dealing with exponential equations as it lets us isolate the exponent, which often contains the variable we are trying to solve for.

Exponential to Logarithmic Form

Translating between exponential and logarithmic forms is a foundational skill when solving exponential equations. As explained earlier, the natural logarithm is the inverse of raising a number to the power of e.

To grasp this concept, it's essential to remember the basic logarithmic identity: a logarithm answers the question 'to what power must we raise the base to obtain a certain number?' Using this, for an equation of the form e^x = y, we can transform it to logarithmic form by taking the natural logarithm of both sides, resulting in ln(e^x) = ln(y), which simplifies to x = ln(y) since ln(e) equals 1. This technique is valuable not just for solving equations but also for understanding the behavior of logarithmic and exponential functions in relation to each other.

Calculator Decimal Approximation

Upon finding the solution to an equation in terms of natural logarithms, we often need a numerical value to interpret our results in a practical context. Calculators come to the rescue at this stage as they can efficiently compute the decimal approximation of expressions involving natural logarithms.

After solving the equation symbolically, you input the logarithmic expression into the calculator. For the given equation in the exercise, you would enter (−ln(793) − 1) / 5. Modern calculators instantly provide an approximate decimal value, which you then round to the required number of decimal places. In many scenarios, like when measuring real-world quantities, this decimal approximation is necessary for reporting a tangible outcome. Despite the calculator's convenience, understanding the underlying mathematics provides deeper insight and a solid foundation for accurately interpreting these decimal results.