Problem 29 - Precalculus with Limits

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(e^{3 x}=12\)

Step-by-Step Solution

Answer

The solution to the equation \(e^{3 x}=12\) is approximately \(x \approx 0.891\)

Step 1: Taking the Natural Logarithm of both sides

Take the natural logarithm (ln) of both sides. This gives you: \(\ln(e^{3x}) = \ln(12)\)

Step 2: Using the properties of logarithms

Use the properties of logarithms to bring the exponent out front: \(3x*\ln(e) = \ln(12)\). This simplifies to \(3x = \ln(12)\)

Step 3: Solving for \(x\)

Divide by 3 on both sides to solve for \(x\): \(x = \ln(12)/3\)

Step 4: Approximating the answer

Calculate the natural logarithm of 12 and divide the result by 3, rounding to three decimal places to get the answer: \(x \approx 0.891\)

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function that is symbolized by 'ln'. It is the inverse operation of taking the exponential of a number with base 'e', where 'e' is the irrational and transcendental constant approximately equal to 2.71828. When you take the natural logarithm of the exponential function, such as (ln(e^x)), you are essentially asking 'to what power do we raise e to get the number x?'. The natural logarithm has a unique property where (ln(e^x) = x), because e raised to the power of x simply returns you back to x. This property is incredibly useful when it comes to solving exponential equations, as it allows us to isolate the exponent for easier calculations.

Properties of Logarithms

Logarithms, including the natural logarithm, have several key properties that make them valuable in algebra. One such property is the Power Rule, which allows you to bring down exponents (such as (ln(a^b) = b*ln(a))). This makes it much easier to handle equations involving e or other bases raised to an unknown power. A few other properties include the Product Rule ((ln(ab) = ln(a) + ln(b))) and the Quotient Rule ((ln(a/b) = ln(a) - ln(b))). Understanding these properties is essential in transforming logarithmic equations into simpler forms that can be easily solved, thereby empowering students to tackle a wide array of algebraic challenges.

Solving an Exponential Equation Algebraically

When faced with an exponential equation where the variable is in the exponent, the natural logarithm is often the key to finding a solution. By taking the natural logarithm of both sides of the equation, you can apply the properties of logarithms to manipulate the equation and isolate the variable. In the given problem, after taking the logarithm on both sides, the properties of logarithms allow us to move the variable x from the exponent to the front, making the equation linear and straightforward to solve. Once the equation becomes (3x = ln(12)), dividing both sides by 3 yields (x = ln(12)/3), which is much simpler to manage and sets the stage for finding the approximate solution.

Approximating Solutions

In algebra, we often seek precise solutions, but in the real world, an approximate answer is sometimes more useful, especially when dealing with irrational numbers that cannot be easily expressed in decimal form. To approximate solutions, a calculator or computer can be used to evaluate expressions like the natural logarithm of a number. After finding the precise value, it can be rounded to a specified number of decimal places for simplicity and practical use. For instance, in the example problem, once x is isolated algebraically, you would use a calculator to compute (ln(12)) and then divide by 3 to obtain the approximate value for x, which in this case is roughly 0.891 when rounded to three decimal places.