Problem 309 - Precalculus

Question

Use logarithms to solve. \(-6 e^{9 x+8}+2=-74\)

Step-by-Step Solution

Answer

The solution is approximately \(x \approx -0.6127\).

Step 1: Isolate the Exponential Term

Start by moving 2 to the other side of the equation to isolate the exponential term:\[-6e^{9x+8}=-74-2\]\[-6e^{9x+8}=-76\]Next, divide both sides by -6 to simplify the expression:\[e^{9x+8}=\frac{76}{6}\]\[e^{9x+8}=\frac{38}{3}\]

Step 2: Apply the Natural Logarithm

Take the natural logarithm of both sides to get rid of the exponential function:\[\ln(e^{9x+8}) = \ln\left(\frac{38}{3}\right)\]The left side simplifies using the property \(\ln(e^y) = y\):\[9x + 8 = \ln\left(\frac{38}{3}\right)\]

Step 3: Solve for x

Now, solve for \(x\) by first subtracting 8 from both sides:\[9x = \ln\left(\frac{38}{3}\right) - 8\]Then, divide by 9 to isolate \(x\):\[x = \frac{\ln\left(\frac{38}{3}\right) - 8}{9}\]

Step 4: Compute the Result

Use a calculator to find \(\ln\left(\frac{38}{3}\right)\) and then substitute the value back to compute \(x\). The approximate value of \(\ln\left(\frac{38}{3}\right)\) is 2.4849. Substitute and solve:\[x = \frac{2.4849 - 8}{9}\]\[x \approx \frac{-5.5151}{9} \approx -0.6127\]

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that can quickly grow or decay. They usually have the form \( f(x) = a \,e^{bx} \) where \( e \) is the base of the natural logarithm, approximately equal to 2.718, and \( a \) and \( b \) are constants. The exponential function is unique because its rate of change is proportional to the function’s own value. This means that the graph of an exponential function can grow quite steeply or decrease toward zero very rapidly, depending on the sign of \( b \).Understanding exponential functions is crucial in many fields, including finance and science, because they describe processes like interest rate growth and radioactive decay. In practical terms, when solving equations with exponential functions, one often needs to isolate the exponential term and then apply logarithms to solve for the variable.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base \( e \). It's the inverse operation of the exponential function with the base \( e \), meaning it can be used to 'undo' the effect of an exponential operation. For example, if you have \( e^y = x \), taking the natural logarithm of both sides will give \( y = \ln(x) \).Using the natural logarithm is a powerful tool for solving exponential equations. It allows us to turn multiplicative solutions into additive problems, making complex calculations simpler. For instance, in the example equation, after isolating the exponential term, taking \( \ln \) on both sides helps us solve for the variable inside the exponent. Remembering the property \( \ln(e^y) = y \) can simplify many problems, converting the exponent into a linear form that can easily be worked with.

Solving Equations

Solving equations, particularly those involving exponential functions, often requires isolating the desired variable. The process generally involves:

• Isolating the exponential part of the equation.
• Taking the natural logarithm of both sides of the equation to deal with the exponential expression.
• Simplifying the expression to solve for the variable.

In the original exercise, isolation was achieved by first moving constants from one side of the equation to another and then dividing by coefficients. Taking the logarithm simplified the exponential component, allowing us to solve the equation like a linear equation. It’s important to be systematic and careful with each step to ensure accuracy and correctness, especially when dealing with equations that involve multiple operations.

Mathematical Properties

Mathematical properties such as logarithmic and exponential identities are essential for simplifying and solving equations. Here are a few useful properties:

• \( \ln(e^y) = y \)
• The logarithm of a quotient: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• The property of inverses: Exponential and logarithm operations cancel each other out.

These properties allow mathematicians to transform challenging exponential equations into manageable algebraic expressions. Recognizing and appropriately using these properties can simplify equations and make problem-solving more efficient. Additionally, in solving logarithmic equations, properties such as these enable converting them into linear forms, which are generally easier to manage and compute.