Problem 43 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln (3 x-4)=2.3 $$

Step-by-Step Solution

Answer

Exact solution: \( x = \frac{e^{2.3} + 4}{3} \); Approximation: \( x \approx 4.6581 \).

Step 1: Isolate the exponential

First, we need to isolate the exponential expression. In this case, we need to solve for the equation inside the natural logarithm. The equation given is:\[ \ln(3x - 4) = 2.3 \]To get rid of the natural logarithm \( \ln \), we raise \( e \) to the power of both sides of the equation to eliminate the \( \ln \). This simplifies our equation to:\[ 3x - 4 = e^{2.3} \]

Step 2: Evaluate the exponential

We evaluate \( e^{2.3} \) using approximate calculations or a calculator to get its value:\[ e^{2.3} \approx 9.9742 \]

Step 3: Solve for x

Having calculated \( e^{2.3} \) as approximately 9.9742, we plug this back into the equation:\[ 3x - 4 = 9.9742 \]Now, solve for \( x \) by first adding 4 to both sides:\[ 3x = 9.9742 + 4 = 13.9742 \]Then divide both sides by 3 to isolate \( x \):\[ x = \frac{13.9742}{3} \approx 4.6581 \]

Step 4: Review and Approximate

The exact solution involves dealing with \( e^{2.3} \), but since its precise calculation provides an irrational number, the approximate solution of \( x \approx 4.6581 \) should be sufficient for practical purposes. Double-check calculations with a calculator, especially for the exponent and division.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), an irrational constant approximately equal to 2.71828. It's widely used in mathematics, especially in calculus and exponential growth problems. The primary role of \( \ln \) is to allow functions involving exponential growth or decay to be linearized, which simplifies many complex mathematical problems.

• Used to solve equations with an unknown inside a logarithmic function.
• \( \ln(e^x) = x \) is a key identity, showing that exponential and logarithmic functions are inverses.
• Helps in converting multiplicative relationships into additive ones.

In the given problem, \( \ln(3x - 4) = 2.3 \), the role of the natural logarithm is to simplify the equation by converting it into an easier-to-solve form with the help of the identity \( e^{\ln(a)} = a \). By applying the exponential function to both sides of the equation, we eliminate the logarithm and directly deal with the relationship expressed inside it.

Exponential Functions

Exponential functions are of the form \( f(x) = e^x \), where \( e \) is the Euler's number. These functions model many natural phenomena like population growth, radioactive decay, and interest calculations in finance. Understanding how to manipulate exponential functions is crucial because they often appear in the context of differential equations and various applied mathematics fields.

• They exhibit rapid growth or decay. This characteristic makes them useful for modeling phenomena with constant relative growth rates.
• The inverse of the natural logarithm \( \ln(x) \) is the exponential function \( e^x \).
• Pressing the exponential button on a scientific calculator usually labeled as "e^x" gives immediate value of exponential function.

In the context of the exercise \( 3x - 4 = e^{2.3} \), calculating \( e^{2.3} \) involves finding the value of this expression, which is approximately 9.9742. The exponential function converts the logarithmic problem into a regular algebraic equation, significantly simplifying the solution process.

Approximate Solutions

When solving equations, an exact answer isn't always available or practical, especially if it involves irrational numbers. Approximate solutions provide a practical approach, giving answers close enough to the exact value for practical purposes.

• They're useful when calculators yield irrational numbers that cannot be expressed precisely.
• Approximations, like rounding to four decimal places, strike a balance between accuracy and simplicity.
• They make computation feasible in real-world applications where slight deviations are acceptable.

In our problem, calculating \( e^{2.3} \) gives us the exact value but simplifying it to around 9.9742 helps avoid errors in further calculations. The approximate solution for \( x \) as 4.6581 demonstrates how effectively approximation complements exact values in practical scenarios. Always ensure that calculations, especially for exponential functions, are double-checked for accuracy.