Problem 17 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(5^{-x / 100}=2\)

Step-by-Step Solution

Answer

The solution is \(x \approx -43.0803\).

Step 1: Understand the Problem

We are given an exponential equation that we need to solve for \(x\). The equation is \(5^{-x / 100} = 2\). Our goal is to find the value of \(x\) that satisfies this equation, rounding the answer to four decimal places.

Step 2: Rewrite the Equation with Logarithms

Since the equation is an exponential form, we can use logarithms to solve it. Take the logarithm of both sides of the equation. Usually, the natural logarithm (ln) or the base-10 logarithm (log) can be used. For this exercise, we'll use the natural logarithm: \[ \ln(5^{-x/100}) = \ln(2). \]

Step 3: Apply Logarithmic Identity

Use the logarithmic identity \(\ln(a^b) = b \ln(a)\) to simplify the equation. This transforms our equation to: \(-\frac{x}{100} \ln(5) = \ln(2).\)

Step 4: Solve for \(x\)

To solve for \(x\), first isolate \(x\) on one side of the equation: \(-\frac{x}{100} = \frac{\ln(2)}{\ln(5)}.\) Multiply both sides by \(-100\) to get \(x\) alone: \[ x = -100 \times \frac{\ln(2)}{\ln(5)}. \]

Step 5: Calculate the Numerical Value

Use a calculator to find the value of \( \ln(2) \) and \( \ln(5) \). Then compute the expression. \( \ln(2) \approx 0.6931 \) and \( \ln(5) \approx 1.6094 \). Thus, \( x = -100 \times \frac{0.6931}{1.6094} \approx -43.0803. \)

Step 6: Round the Answer

The problem requires us to round the solution to four decimal places. Therefore, the final answer is \(-43.0803\).

Key Concepts

Natural Logarithms

Natural logarithms, denoted as \( \ln(x) \), are a specific type of logarithm that use the constant \( e \) (approximately 2.718) as their base. Logarithms are a way of expressing numbers as powers or exponents, and the natural logarithm is particularly useful in calculus and many real-world applications. In this exercise, we used natural logarithms to solve an exponential equation. By applying the \( \ln \) function to both sides of an equation, we can transform an exponential equation into a linear one, which is easier to calculate and solve.

• The key property of the natural logarithm is that \( \ln(e^x) = x \), serving as the inverse function of the exponential function \( e^x \).
• Natural logarithms appear frequently in equations involving growth processes, such as population growth or radioactive decay, due to their exponential nature.

In our problem, using natural logarithms allowed us to unlock the power of mathematical identities to simplify and solve the exponential equation.

Logarithmic Identities

Logarithmic identities are mathematical tools that help simplify and solve logarithmic and exponential equations. One key identity is \( \ln(a^b) = b \ln(a) \), which allows us to bring the exponent in a logarithmic expression down as a multiplier. This identity was crucial in solving our exercise, allowing us to transform \( \ln(5^{-x/100}) \) into \(-\frac{x}{100} \ln(5) \).

• Another important identity is \( \ln(ab) = \ln(a) + \ln(b) \), useful for breaking down products within logarithmic terms.
• There's also \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \), which helps isolate terms within fractions.

These identities are essential for students to learn as they unlock the ability to simplify complex logarithmic expressions and tackle a variety of problems. By mastering these, students can handle both straightforward calculations and more complex logarithmic transformations.

Solving Equations

Solving equations, especially those involving exponential and logarithmic forms, requires a systematic approach. Starting with understanding the equation, we identify the type of equation we are dealing with — in this case, exponential. We then choose an appropriate method to transform it into a form that's easier to work with. For our problem, this transformation was achieved by using logarithms.

Once transformed, we applied algebraic techniques to isolate the variable of interest, \( x \). The logarithmic identities allowed us to simplify expressions along the way. In our exercise, multiplying by \(-100\) subsequently isolated \( x \), allowing us to solve step by step until we reached the solution.

• It's pivotal to check the final solution for precision by reviewing each step and performing accurate calculations, especially with logarithms that require calculator inputs.
• Rounding correctly as per the problem's instructions ensures that the answer meets the precision required, as seen in rounding \(-43.0803\) to four decimal places.

Practicing these strategies across various problems helps students gain confidence and proficiency in solving not just exponential, but a wide range of mathematical equations.