Problem 11 - Calculus

Question

Solve for \(x\) using logs. $$7=5 e^{0.2 x}$$

Step-by-Step Solution

Answer

x ≈ 1.6825

Step 1: Isolate the Exponential Term

The given equation is \(7 = 5 e^{0.2x}\). To isolate the exponential term, divide both sides of the equation by 5:\[ \frac{7}{5} = e^{0.2x} \]

Step 2: Apply Natural Logarithm

To solve for \(x\), apply the natural logarithm to both sides of the equation. The natural log will cancel out the exponential on the right side of the equation:\[ \ln\left(\frac{7}{5}\right) = \ln(e^{0.2x}) \]

Step 3: Simplify Using Logarithmic Rules

Using the logarithmic identity \(\ln(a^b) = b\ln(a)\), the equation simplifies to:\[ \ln\left(\frac{7}{5}\right) = 0.2x \]

Step 4: Solve for x

To find \(x\), divide both sides of the equation by 0.2:\[ x = \frac{\ln\left(\frac{7}{5}\right)}{0.2} \]

Step 5: Compute the Result Numerically

Calculate the right side to find the numerical value of \(x\):\[ x \approx \frac{\ln(1.4)}{0.2} \approx \frac{0.3365}{0.2} \approx 1.6825 \]

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is represented in the form:

• \(f(x) = a^{x}\)

where \(a\) is a constant, and \(x\) is the variable exponent. These functions show exponential growth or decay depending on the base value and the exponent.
Exponential functions are important because they model a wide range of real-world situations, such as population growth, radioactive decay, and compound interest.
In our exercise example, the exponential function is \(e^{0.2x}\), where \(e\) is the base of the natural logarithm, approximately 2.718. The exponent is a linear expression, \(0.2x\), which suggests a continuous growth rate applied to the variable \(x\). By solving exponential equations, we often aim to find the value of \(x\) that satisfies specific conditions, as shown in the exercise.

Natural Logarithm

The natural logarithm, often denoted by \(\ln(x)\), is a logarithm with the base \(e\), where \(e\) is Euler's number, approximately equal to 2.718. It is the inverse operation of the exponential function, meaning it can "undo" exponentiation.
For example, if \(e^y = x\), then \(\ln(x) = y\). This property makes the natural logarithm particularly valuable for solving equations involving exponential terms.
In our step-by-step solution, we applied the natural logarithm to both sides of the equation to solve for \(x\). By writing \(\ln(e^{0.2x})\), we were able to utilize the property that the logarithm of an exponential expression is just the exponent, \(0.2x\). Therefore, this step simplifies our equation significantly, letting us isolate and solve for \(x\).

Logarithmic Identities

Logarithmic identities are useful mathematical properties that simplify the manipulation of logarithmic expressions. Understanding these identities is crucial for solving logarithmic and exponential equations. Some key identities include:

• \(\ln(a^b) = b\ln(a)\)
• \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\)
• \(\ln(xy) = \ln(x) + \ln(y)\)

In the exercise solution, we used the identity \(\ln(a^b) = b\ln(a)\), which allows us to bring the exponent in front of the logarithm as a multiplier. This identity helped simplify \(\ln(e^{0.2x})\) to \(0.2x\ln(e)\). Given that \(\ln(e) = 1\), our equation further reduces to \(0.2x\).
Through these identities, logarithms become more manageable, enabling us to isolate variables and solve equations efficiently. The power of logarithmic identities lies in their ability to transform complex expressions into simpler, solvable forms.