Problem 60 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$2 \ln x-\ln 5=\ln (x+10)$$

Step-by-Step Solution

Answer

The short answer version of solving the logarithmic equation \(2 \ln x-\ln 5=\ln (x+10)\) is to first simplify the equation using logarithmic properties: \[\ln{\left(\frac{x^2}{5}\right)}=\ln{(x+10)}\] Then, transform and solve the equation: \[\frac{x^2}{5}=x+10 \implies x^2-5x-50=0 \implies (x-10)(x+5)=0\] The solution is \(x=10\) since \(x=-5\) is an invalid solution for natural logarithms. To verify the solution, use a graphing calculator to graph the functions \(y_1=2 \ln x-\ln 5\) and \(y_2=\ln (x+10)\), and observe an intersection point at (10, 4.605), confirming the solution is correct.

Step 1: Simplify the equation using logarithmic properties

We have the equation \(2 \ln x-\ln 5=\ln(x+10)\). First, use the logarithmic property \(\ln{a^n} = n\ln{a}\) to rewrite the given equation: \[\ln{(x^2)}-\ln{5}=\ln{(x+10)}\] Now, use the logarithmic property \(\ln{a}-\ln{b}=\ln{\frac{a}{b}}\) to combine the logarithmic terms on the left side of the equation: \[\ln{\left(\frac{x^2}{5}\right)}=\ln{(x+10)}\]

Step 2: Transform the equation

Since we have the equation \(\ln{\left(\frac{x^2}{5}\right)} = \ln{(x+10)}\), we can eliminate the logarithms by using the property \(\ln{a}=\ln{b} \implies a=b\): \[\frac{x^2}{5}=x+10\]

Step 3: Solve the equation

Now, we have a quadratic equation by multiplying both sides by 5: \[x^2 = 5(x+10)\] Expand the equation: \[x^2 = 5x+50\] Move all the terms to one side to get the equation into standard form: \[x^2 - 5x - 50 = 0\] Factor the quadratic equation: \[(x-10)(x+5) = 0\] Now, find the solutions for x: \(x-10=0 \implies x=10\) \(x+5=0 \implies x=-5\) However, natural logarithms can only have positive arguments, so x = -5 is an invalid solution. Thus, the only valid solution is: \[x=10\]

Step 4: Verify the solution using a graphing calculator

To check the solution, graph the following functions on a graphing calculator: \(y_1=2 \ln x-\ln 5\) \(y_2=\ln (x+10)\) Observe where the graphs intersect. This will be the solution to the logarithmic equation. The intersection point observed is at (10, 4.605), which verifies that our algebraic solution \(x=10\) is correct.

Key Concepts

Understanding Logarithmic Properties

Logarithmic properties can be incredibly useful when solving equations that involve logarithms, such as those with natural logarithms. There are several key properties of logarithms that simplify equations:

• The power rule: \(\ln{a^n} = n \ln{a}\). This property helps to bring variables in exponents down to a more manageable form.
• The quotient rule: \(\ln{a} - \ln{b} = \ln{\frac{a}{b}}\). **It allows us to combine or separate logarithms into a single expression.**
• The rule of equality: **\(\ln{a} = \ln{b} \implies a = b\).** If two logarithms are equal, then their arguments are equal as well.

These properties are incredibly useful in simplifying complex logarithmic expressions, making equations easier to solve.** When dealing with logarithms, breaking them down using these rules can turn a complicated problem into a simpler mathematical expression.** Always make sure your arguments are positive, as logarithms are undefined for non-positive numbers.

Breaking Down Quadratic Equations

Quadratic equations are polynomials of degree two and can be written in the standard form of \(ax^2 + bx + c = 0\). Solving these equations often involves factoring, using the quadratic formula, or by completing the square.

• **Factoring:** When the quadratic equation can be expressed as a product of two binomials, it is possible to quickly find the solutions by setting each binomial equal to zero.
• **Completing the Square:** This method revises the original equation into a perfect square trinomial form, making it straightforward to solve.
• **Quadratic Formula:** Useful when the equation cannot be easily factored, this formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) is derived from the process of completing the square and works for any quadratic equation.

In the context of the original exercise, after simplifying the logarithmic equation, it transitioned into a quadratic form. Factoring was employed to solve for the variable, but always check your answers to make sure they fit any initial conditions, especially when dealing with logarithms.

An Introduction to Natural Logarithms

Natural logarithms are a type of logarithm with the base \(e\), which is approximately equal to 2.718, known as Euler's number. **The notation used is \(\ln{x}\) rather than the usual logarithmic notation of \(\log_b{x}\) for other bases.**

• Natural logarithms play a crucial role in various fields, including calculus, complex numbers, and growth models because they simplify the process of solving exponential equations involving the base \(e\).
• They are inverse functions of the exponential function \(e^x\), meaning \(\ln(e^x) = x\) and \(e^{\ln{x}} = x\), which is very helpful in solving equations where the variable is an exponent.
• Natural logs adhere to the same logarithmic properties as other bases, which makes them both versatile and powerful for algebraic manipulation.

In our problem, we used natural logarithms and applied the logarithmic properties to simplify and solve a more complex logarithmic equation algebraically. Being comfortable with natural logs is essential for solving higher-level math problems.

Utilizing Graphing Calculators for Verification

Graphing calculators are powerful tools that can help verify the solutions of equations visually. **They plot graphs to find where functions intersect, revealing solutions that might otherwise be tedious to solve algebraically.**

• By entering equations into a graphing calculator, you can quickly visualize where graphs of expressions meet, which corresponds to the solutions of the equation.
• They can also be used to confirm the behavior of the function across different values, which is especially useful for complicated logarithmic and exponential equations.
• Graphing various functions together can provide a visual confirmation that complements algebras' numerical methods.

In the exercise, the algebraic solution \(x = 10\) was verified by graphing the given functions on a calculator. Observing the intersection at \(x=10\) helped confirm the correctness of the algebraic process. **Using such technology can bridge the gap between abstract algebraic equations and tangible visual solutions**.