Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3(1.4)^{x}-4=60$$

Step-by-Step Solution

Answer

The solution is approximately \( x = 9.107 \).

Step 1: Isolate the Exponential Expression

Start with the equation:\[ 3(1.4)^x - 4 = 60 \]Add 4 to both sides of the equation to isolate the term with the exponential expression:\[ 3(1.4)^x = 64 \]

Step 2: Divide to Simplify

Divide both sides by 3 to solve for the exponential term:\[ (1.4)^x = \frac{64}{3} \]This simplifies the equation to:\[ (1.4)^x \approx 21.333 \]

Step 3: Take the Logarithm of Both Sides

To solve for \( x \), take the logarithm (using any logarithm base, but typically base 10 or natural log) of both sides:\[ \log((1.4)^x) = \log(21.333) \]Using the logarithmic identity \( \log(a^b) = b \log(a) \), we have:\[ x \log(1.4) = \log(21.333) \]

Step 4: Solve for \( x \)

Divide both sides by \( \log(1.4) \) to solve for \( x \):\[ x = \frac{\log(21.333)}{\log(1.4)} \]

Step 5: Calculate the Exact Solution

Calculate the exact value of \( x \) using a calculator:\( x \approx \frac{1.329}{0.146} \)Calculate this value to find:\( x \approx 9.1 \)

Step 6: Round to the Nearest Thousandth

Since you need the solution accurate to the nearest thousandth, round the result: \[ x \approx 9.107 \]

Key Concepts

Logarithms

Logarithms are incredibly useful in solving exponential equations. They transform the multiplicative form of an equation into an additive one, which is much simpler to manage. In our exercise, we moved from needing to solve \((1.4)^x = 21.333\) by using logarithms to find the unknown exponent.

• The concept of logarithms revolves around identifying the power to which a base number must be raised to produce a given number. For instance, log(base 10) of 100 is 2, because 10 raised to the power of 2 is 100.
• We use the identity \(\log(a^b) = b \times \log(a)\) to bring the exponent down into a form we can handle. This lets us solve for \(x\) algebraically.
• Logarithms can be of any base, but base 10 (common log) and base \(e\) (natural log) see the most use in calculations.

Understanding logarithms expands our mathematical toolkit, making it possible to solve complex exponential equations, just like in our example problem.

Rational and Irrational Numbers

In mathematics, differentiating between rational and irrational numbers is vital when expressing solutions. Rational numbers can be represented as fractions or ratios of integers. For example, \(64/3\) is a rational number since it is a simple fraction.
However, when calculating or approximating powers and logarithms, we often encounter irrational numbers. These cannot be expressed as perfect fractions and have non-repeating, non-terminating decimals. Much like our calculated value of \(x \approx 9.107\) in the exercise.

• Every rational number can be written in the form \(p/q\), where \(p\) and \(q\) are integers and \(q eq 0\).
• Irrational numbers cannot be precisely written as fractions. Examples include \(\sqrt{2}\), \(\pi\), and numbers such as \(9.107\) which result from solving complex exponential equations without a repeating pattern.
• When an exact solution is irrational, rounding to a certain decimal place, such as the nearest thousandth, helps provide a usable approximation.

By understanding these types of numbers, we can comfortably express solutions with the appropriate level of precision.

Calculator Use

Calculators make solving exponential equations much more manageable, especially when dealing with complex values involving logarithms. They provide numerical solutions that might be cumbersome to compute by hand. For solving an equation like \(3(1.4)^x - 4 = 60\), a calculator is indispensable for simplifying and approximating calculations.

• Use the calculator to find logarithms by entering the right functions, like log base 10 or natural log, exactly as needed.
• For precise results, ensure the calculators are set to calculate to at least three decimal places when solving for \(x\) to get values such as \(\log(21.333)\).
• Check that your calculator's mode is suitable for the equation; typically, this means staying in a standard decimal mode rather than binary or hexadecimal.

Leveraging a calculator's power enables efficient and accurate processing of complex mathematics beyond basic arithmetic. With technology at our fingertips, we can solve intricate logarithmic transformations and exponential equations swiftly and accurately.