Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(7+3 \ln x=5\)

Short answer

The solution to the equation is \(x \approx 0.513\)

Step-by-step solution

Isolate the logarithm

First, subtract 7 from both sides of the equation to get the logarithmic part of the equation by itself. This gives us: \(3 \ln x = 5 - 7, 3 \ln x = -2\)

Eliminate the coefficient of the logarithm

To remove the 3, which is the coefficient of the logarithm, divide both sides of the equation by 3. This provides \( \ln x = -2/3 = -0.667\).

Solve for x

We apply the definition of a logarithm to solve for \(x\). We use the fact that if \(a = \ln b\), then \(b = e^a\), where e is approximately 2.71828. Hence \(x = e^{-0.667}\).

Approximate x to three decimal places

Use a calculator to find the approximate value of \(x\). This gives \(x \approx 0.513\) when rounded to three decimal places.

Key concepts

Logarithmic Equation Algebra

Solving logarithmic equations algebraically requires an understanding of how to manipulate and solve equations that involve logarithms. The key to successfully navigating logarithmic equations is to isolate the logarithmic term. This typically involves using basic algebraic principles like moving terms across the equality sign and applying inverse operations.

For example, in the given exercise, we start by moving the non-logarithmic term to the other side of the equation to focus on the logarithmic part. This step is crucial because it simplifies the equation to a form where the logarithmic properties can be applied more straightforwardly. In essence, algebra with logarithms isn't much different from regular algebra — it's about isolating the variable and simplifying the terms, but with an extra layer of understanding logarithmic rules.

Isolating the Logarithm

Isolating the logarithm is a pivotal technique in solving logarithmic equations. To isolate the logarithm, we may need to perform operations like addition or subtraction to remove any constants attached to the logarithmic term, and division or multiplication to remove any coefficients.

As seen in the exercise, subtracting 7 from both sides, followed by dividing by 3, leaves the logarithm by itself on one side of the equation. Only when the logarithm is isolated can we effectively apply exponential functions to both sides to solve for the variable within the logarithm, which ultimately allows us to find the value of x in its exponential form.

Logarithm Properties

Understanding logarithm properties is essential when working with logarithmic equations. One of the most important properties is the definition, which states that if \(a = \log_b(c)\), then \(b^a = c\). This property allows us to rewrite logarithms as exponential expressions and vice versa, which is the fundamental concept behind solving for x in logarithmic equations.

Other logarithm properties include the product rule, quotient rule, and power rule, which allow us to manipulate logarithmic expressions in ways that make them easier to solve. For instance, the power rule, which states that \(\log_b(c^a) = a\log_b(c)\), can be applied inversely to take a coefficient of a logarithm and turn it into an exponent within the logarithm, further helping to isolate and solve for the variable.

Exponential Functions

Exponential functions are the inverse of logarithms and are used to solve for the variable once the logarithm is isolated. If \(a = \ln(b)\), then by definition \(b = e^a\), where \(e\) is the base of the natural logarithm, an irrational constant approximately equal to 2.71828.

In the step-by-step solution provided, once the logarithm was isolated, the equation looked like \(\ln(x) = -0.667\). By applying the definition of the natural logarithm, we get \(x = e^{-0.667}\). This step is where the transition from logarithmic to exponential form happens. A calculator is then used to approximate the value of x to three decimal places, which gives us a tangible number we can use in practical scenarios.