Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(2\left(3^{x}\right)=16\)

Step-by-Step Solution

Answer

The solution is \(x \approx 1.893\)

Step 1: Isolate the exponential expression

In order to isolate the exponential part \(3^{x}\), the equation \(2(3^{x}) = 16\) should be divided by 2 on both sides. This will result in \(3^{x} = 8\).

Step 2: Use logarithms to deal with the exponent

The unknown exponential power can be eliminated by taking the natural log on both sides of the equation, leading to \(\ln(3^{x}) = \ln(8)\). By properties of logarithms this simplifies to \(x\ln(3) = \ln(8)\).

Step 3: Solve for x

To isolate x, divide both sides of the equation \(x\ln(3) = \ln(8)\) by \(\ln(3)\) to obtain \(x = \frac{\ln(8)}{\ln(3)}\).

Step 4: Approximate the result

Using a calculator the decimals of x can be approximated to three places leading to \(x \approx 1.893\).

Key Concepts

Isolate the Exponential Expression

Understanding how to isolate the exponential expression is the initial and crucial step in solving exponential equations. In the exercise, the equation is given as \(2(3^{x}) = 16\). By dividing both sides by 2, we simplify the equation to obtain \(3^{x} = 8\). This step clears the path to work directly with the exponential component without any further distractions. Simplifying an equation at an early stage can prevent mistakes during the more complex steps that follow, such as applying logarithms.

Use Logarithms

Once the exponential expression is isolated, the subsequent move involves utilizing logarithms. Logarithms are the inverse function of exponentiation and are particularly wieldable for handling equations where the variable is an exponent. Our example equation, \(3^{x} = 8\), becomes \(\ln(3^{x}) = \ln(8)\) when we apply the natural logarithm on both sides. This transformation strategically sets the stage for the application of properties of logarithms, allowing us to untangle the exponent from its base.

Properties of Logarithms

Leveraging the properties of logarithms is essential in solving equations with exponents. One of the core properties used here is the power rule, which states that \(\ln(a^{n}) = n\ln(a)\). Thus, we can rewrite our example as \(x\ln(3) = \ln(8)\), effectively moving the exponent outside the logarithm. This step drastically simplifies our task, transforming an exponential problem into a much more manageable linear one. Furthermore, knowing other properties such as the product, quotient, and change of base rules can greatly expand your problem-solving toolkit.

Approximate Decimal Results

To approximate decimal results of a logarithmic equation, one typically resorts to a scientific calculator due to the non-elementary nature of most logarithmic values. Following the isolation of the variable 'x', we use the ratio \(\frac{\ln(8)}{\ln(3)}\) to calculate its numerical value. The result is a transcendental number and must be rounded for practical use. In the question, it is specified to round the answer to three decimal places, yielding \(x \approx 1.893\). This process provides not only an approximation suitable for real-world application but also helps in understanding the sensitivity of exponential equations to their components.