Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Round your result to three decimal places. $$7 \ln 3 x=12$$

Step-by-Step Solution

Answer

Therefore, rounding to three decimal places, the solution for x in the equation \(7 \ln 3 x = 12\) is \(x = 2.120\).

Step 1: Isolate the logarithm

To do this, divide both sides of the equation by 7. This gives:\[\ln(3x)= \frac{12}{7}\]

Step 2: Convert from logarithmic to exponential form

The equation \(\ln(a)=b\) can be converted to an equivalent equation in exponential form, which is \(e^b=a\). So the above equation becomes:\[e^{\frac{12}{7}}=3x\]

Step 3: Solve for x

To solve for x, divide both sides by 3. It gives:\[x = \frac{e^{\frac{12}{7}}}{3}\]

Key Concepts

Exponential Functions

Exponential functions are a crucial part of mathematics, widely used in various fields like finance, engineering, and sciences. These functions have the form \(y = a^x\), where \(a\) is a constant and \(x\) is the exponent. They are characterized by having a constant rate of growth or decay. For instance, in the real world, exponential functions describe population growth, radioactive decay, and interest calculations.

When dealing with equations involving exponential functions, like \(e^b = a\), we often need to convert them from their logarithmic forms. This is possible because exponents and logarithms are inverse operations. This relationship helps in transitioning between logarithmic and exponential forms of equations, which is immensely useful in solving mathematical and real-world problems.

Understanding exponential functions allows you to grasp how quickly values can increase or decrease, based on the exponential rate. This understanding is vital in predicting outcomes and making informed decisions which involve exponential changes.

Solving Equations

Solving equations is a fundamental skill in algebra and mathematics in general. It involves finding the value of the unknown variable that makes the equation true. There are various methods to solve equations, including factoring, using the quadratic formula, or substitution.

In the given problem, the equation involves a natural logarithm, and to solve it, we must first isolate the logarithmic term. To do this, each side of the equation is manipulated, such as dividing by constants or multiplying through by numbers, to simplify and solve for the variable. Once the equation is simplified, it is often necessary to transform it into a different form – for our example, from a logarithmic equation \(\ln(3x) = \frac{12}{7}\) to an exponential form \(e^{\left(\frac{12}{7}\right)} = 3x\), through which the solution becomes more apparent.

Understanding how to systematically solve equations by isolating terms and converting forms is essential for solving complex problems involving various mathematical operations.

Natural Logarithm (ln)

The natural logarithm, represented as \(\ln\), is a special logarithm with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is widely used in calculus and mathematical analysis because of its natural properties in relation to rates of growth and exponential decay.

When solving equations involving \(\ln\), understanding its properties and how it interacts with exponential functions is crucial. The property \(\ln(e^x) = x\) highlights the inverse relationship between natural logarithms and exponential functions. This property allows us to convert a logarithmic equation into an exponential form, simplifying the solving process.

In practical applications, natural logarithms are used to solve problems involving continuous growth or decay, such as in biology for modeling population growth or in finance for computing compound interest. Knowing how to manipulate and solve equations using \(\ln\) enables you to tackle a wide range of logarithmic problems confidently.