Problem 63 - Essential Calculus Early Transcendentals

Question

Solve each equation for \(x\) $$ \text { (a) }e^{7-4 x}=6 \quad \text { (b) } \ln (3 x-10)=2 $$

Step-by-Step Solution

Answer

(a) \(x = \frac{7 - \ln(6)}{4}\), (b) \(x = \frac{e^2 + 10}{3}\).

Step 1: Solve Equation (a)

We need to solve the equation \(e^{7-4x} = 6\) for \(x\). To do this, we start by taking the natural logarithm of both sides:\[\ln(e^{7-4x}) = \ln(6)\]Use the property of logarithms: \(\ln(e^a) = a\). Then the equation becomes:\[7 - 4x = \ln(6)\]Next, solve the equation for \(x\) by isolating it on one side:\[4x = 7 - \ln(6)\]Divide each term by \(4\):\[x = \frac{7 - \ln(6)}{4}\]

Step 2: Solve Equation (b)

Now, solve the equation \(\ln(3x - 10) = 2\). Begin by exponentiating both sides to eliminate the natural logarithm.\[e^{\ln(3x - 10)} = e^2\]Using the property \(e^{\ln(a)} = a\), it simplifies to:\[3x - 10 = e^2\]Then solve for \(x\) by adding 10 to both sides:\[3x = e^2 + 10\]Finally, divide both sides by 3 to isolate \(x\):\[x = \frac{e^2 + 10}{3}\]

Key Concepts

Natural Logarithm

The natural logarithm, often abbreviated as "ln," is a logarithmic function with the base of the mathematical constant e, approximately 2.71828. The natural logarithm is useful in solving equations involving exponential functions, particularly when the base of the exponential is e.

• The natural log of a number x, written as \(\ln(x)\), is the power to which e must be raised to produce x.
• One of the key properties of the natural logarithm is that \(\ln(e^a) = a\), which allows us to simplify expressions where the variable is an exponent of e.
• This property is particularly helpful when solving equations like \(e^{7-4x} = 6\), as we can apply the natural log to both sides to bring the exponent down, making it easier to isolate the variable.

Understanding the natural logarithm is fundamental when dealing with both exponential functions and equations, as it provides a way to "undo" the exponential, allowing for the simplification and subsequent solving of complex equations.

Exponential Equations

Exponential equations are those in which variables appear as exponents. These equations generally take the form \(b^{x} = y\), where b is a positive number, x is the exponent, and y is the output. Solving exponential equations often requires the use of logarithms.

• When the base of the exponent is e, the equation can be simplified using the natural logarithm, as seen in the equation \(e^{7-4x} = 6\).
• By taking \(\ln\) of both sides, we reduce the exponential equation to a linear one, simplifying the process of finding x.
• Alternatively, if the equation involves natural logarithms, you might exponentiate both sides to remove the logarithm, transforming it into a more straightforward form.

Solving exponential equations typically necessitates a solid grasp of algebraic manipulation and the properties of logarithms, allowing for the conversion of complex forms into simpler, more manageable equations.

Solving for Variables

In mathematics, solving for variables involves finding the value of unknowns that satisfy an equation. This process can involve several algebraic techniques and sometimes requires understanding advanced concepts like logarithms or exponents.

• To solve \(e^{7-4x} = 6\), one must first use the natural log to bring the exponent down to the level of regular algebra, resulting in \(7 - 4x = \ln(6)\).
• From here, basic algebraic principles apply, such as isolating the variable by moving terms across the equation and performing arithmetic operations.
• In other cases, like \(\ln(3x - 10) = 2\), exponentiating both sides can eliminate the natural logarithm, simplifying the path to isolating and solving for x.

Each equation might require a different strategy, but the ultimate goal remains consistent: to isolate the variable and solve for its value in a way that satisfies the original equation.