Problem 15 - College Algebra

Question

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7^{x+2}=410$$

Step-by-Step Solution

Answer

\(x \approx 3.55\), correct to two decimal places. Note that decimal approximation might vary slightly depending on the calculator.

Step 1: Write the Equation in Logarithmic Form

First, write the equation in logarithmic form to simplify the equation into a format that will allow us to solve for \(x\). This transition from exponential form to logarithmic form is essential in understanding the movements in the equation. Here it is as a logarithmic equation: \(x + 2 = \log_7 410\).

Step 2: Use the Change of Base Formula

Next, we will use the Change of Base Formula. Since using a base of 7 might not be handy for our calculator, we will change the base to \(e\), which permits us to use natural logarithms. Remember, the Change of Base Formula is: \(\log_b a = \frac{\ln a}{\ln b}\), where \(a\) and \(b\) are the constants in our original equation and “ln” denotes the natural log. So, our equation becomes: \(x + 2 = \frac{\ln 410}{\ln 7}\).

Step 3: Solve for \(x\)

Finally, isolate \(x\) on one side to solve for it. This is obtained by subtracting 2 from both sides of the equation: \(x = \frac{\ln 410}{\ln 7} - 2\). In order to get a decimal approximation of the answer, one can substitute this expression into a calculator.

Key Concepts

Natural Logarithms

Natural logarithms, often represented as 'ln', are a fundamental concept in mathematics, particularly when dealing with exponential equations. These logarithms have a base of Euler's number, 'e', which is approximately 2.71828. This number is significant because it naturally arises in various areas of mathematics, including calculus and complex analysis.

Natural logarithms convert the process of multiplication into addition, and the process of raising to a power into multiplication, which makes calculations with large numbers or complex growth and decay problems more manageable. In the context of solving exponential equations, using natural logarithms can simplify the process since they are the inverse operation of exponentiating with the base 'e'.

When you see an equation set in the form of 'e' to the power of 'x' equals 'y', taking the natural logarithm of both sides will allow you to solve for 'x' because the natural log and 'e' cancel each other out. This property is especially useful when 'y' is not a power of 'e', and we're seeking to express 'x' explicitly.

Change of Base Formula

The Change of Base Formula is a handy tool for solving equations with bases other than 'e' and 10, which are the bases for natural and common logarithms respectively, and allows for easier computation using a standard scientific calculator.

The formula is given as \(\log_b a = \frac{\ln a}{\ln b}\), where 'a' is the number we are taking the logarithm of, 'b' is the base of the logarithm, 'ln' represents the natural logarithm, and '\(\log_b\)' signifies the logarithm with base 'b'. When we apply this formula, we switch from a less common base 'b' to base 'e', because calculating '\(\ln\)' values directly is more practical with modern calculators.

In practice, to solve for an unknown exponent in an exponential equation, we might first rewrite the equation in logarithmic form and then apply the Change of Base Formula to work with natural logarithms. This strategy allows us to solve equations that would otherwise be difficult to calculate directly and is integral in our ability to solve the given textbook exercise.

Logarithmic Form

Understanding the logarithmic form is essential for solving exponential equations. An exponential equation is typically presented in the form '\(b^x = y\)', where 'b' is the base, 'x' is the exponent, and 'y' is the result of the exponential expression.

To solve for the variable exponent, we can translate this exponential equation into its logarithmic form, which looks like '\(x = \log_b y\)'. This transformation leverages the defining property of logarithms: the exponent 'x' is the power to which the base 'b' must be raised to yield the number 'y'. In simpler terms, logarithmic form allows us to 'unpack' the exponent in an equation, making it accessible for isolation and solution.

By transforming the textbook exercise's exponential equation into a logarithmic equation, we easily move from an equation where the unknown is in an exponent (a more complex scenario to handle algebraically) to one where the unknown is the solution to a logarithm (which is more straightforward to isolate and solve). This step is critical for finding the value of 'x' in the provided exercise.