Problem 9 - 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. $$3^{x}=7$$

Step-by-Step Solution

Answer

Exact Solution: \( x = \frac{\ln(7)}{\ln(3)} \). Approximation: \( x \approx 1.771 \).

Step 1: Apply Logarithms to Both Sides

To solve the equation \(3^{x}=7\), apply the natural logarithm (\(\ln\)) to both sides. This allows you to bring down the exponent in front of the logarithm using the power rule of logarithms. The equation becomes \(\ln(3^{x}) = \ln(7)\). By using the power rule, it simplifies to \(x \cdot \ln(3) = \ln(7)\).

Step 2: Solve for x in Exact Form

Next, solve for \(x\) by isolating it. Divide both sides of the equation by \(\ln(3)\) to find \(x = \frac{\ln(7)}{\ln(3)}\). This is the exact form of the solution.

Step 3: Use a Calculator for Approximation

To approximate the solution to the nearest thousandth, use a calculator to find the values of \(\ln(7)\) and \(\ln(3)\). After calculating, \(\ln(7) \approx 1.945\) and \(\ln(3) \approx 1.099\). Substitute these into the expression \(x = \frac{\ln(7)}{\ln(3)}\) to find \(x \approx \frac{1.945}{1.099} \approx 1.771\).

Step 4: Verify the Solution

Ensure that the calculated approximation is correct by substituting \(x \approx 1.771\) back into the original equation and checking if \(3^{1.771} \approx 7\) using a calculator. This reaffirms that the approximated value is accurate to the nearest thousandth.

Key Concepts

Natural Logarithm

Natural logarithms are a special type of logarithm that use the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In many equations like the exponential equation \(3^x = 7\), using natural logarithms can simplify the process of solving for \(x\).

• Why use natural logarithms? Because they make it easier to deal with exponents analytically.
• This is due to the properties of logarithms that allow us to "bring down" the exponent, simplifying the calculations.
• Natural logarithms are represented as \(\ln(x)\).

In this problem, applying \(\ln\) to both sides gives an equation \(\ln(3^x) = \ln(7)\), which can then be manipulated using properties of logarithms, creating manageable algebraic expressions.

Power Rule

The power rule of logarithms is a key concept that helps in solving exponential equations like our example \(3^x = 7\). The rule states that the logarithm of a power is the exponent times the logarithm of the base: \(\ln(a^b) = b \cdot \ln(a)\).

• This property is crucial because it lets us take the exponent out in front, turning a hard-to-solve equation into a much simpler one.
• In our step-by-step solution, after applying the natural logarithm to both sides, we used the power rule to get \(x \cdot \ln(3) = \ln(7)\).
• Doing so transforms the exponential equation into a linear one, which is straightforward to solve.

Hence, thanks to the power rule, finding the value of \(x\) becomes a simple matter of arithmetic.

Exact Form

An exact form solution in mathematics excludes any approximations or rounding. It's particularly common in solving equations where precision is necessary. For the equation \(3^x = 7\), the exact form is represented as \(x = \frac{\ln(7)}{\ln(3)}\).

• Exact solutions are crucial for understanding the precise mathematical relationships and can provide insights into the structure of the solution.
• This form remains untouched by numerical approximation errors.
• It implies that you are utilizing the full precision of mathematical constants like \(\ln(7)\) and \(\ln(3)\) without simplifying them to decimal values.

Understanding and obtaining an exact form ensures that you have a firm grip on the intricate aspects of mathematical relationships.

Approximation

Approximations offer a practical way to express solutions when dealing with irrational numbers or complex calculations in everyday situations. When solving \(3^x = 7\), we eventually want a more digestible result, which we achieve through approximation.

• The process involves using calculators to determine the numerical value of logarithms to a desired degree of accuracy.
• In this specific problem, we approximate \(\ln(7) \approx 1.945\) and \(\ln(3) \approx 1.099\).
• These values help us approximate \(x\) by performing the division \(x \approx \frac{1.945}{1.099} \approx 1.771\).

Approximation is useful not only for making sense of an answer but also for checking the plausibility of results in both academic and real-world contexts.