Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of 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

The solution to the equation \(7^{x+2}=410\) is \( x = 3.97 \) after approximated to two decimal places.

Step 1: Write the Original Equation

The original equation is \(7^{x+2}=410\).

Step 2: Apply Logarithm to Both Sides

Apply a logarithm to both sides of the equation. For this example, we'll use the natural logarithm (ln). This gives us: \(\ln(7^{x+2}) = \ln(410)\).

Step 3: Use Logarithm Property

Use the property of logarithms \( \log_b(a^n) = n \log_b(a) \). This allows you to pull the exponent out in front of the logarithm. So the equation becomes: \( (x + 2)\ln(7) = \ln(410) \).

Step 4: Isolate x

Isolate the unknown x by subtracting 2 from both sides and then dividing both sides by \( \ln(7) \). This leads to the equation: \( x = \frac{\ln(410)}{\ln(7)} - 2 \).

Step 5: Solve for x

Now, use a calculator to find the approximate value for \( x \) correct to two decimal places. This will give us \( x = 3.97 \).

Key Concepts

Logarithmic Transformation

Understanding logarithmic transformation is essential when it comes to solving exponential equations. This process begins with an equation such as \(7^{x+2}=410\) and converts it into a form that is easier to work with using properties of logarithms. Logarithmic transformation gives us a way to deal with the exponent by applying a logarithm to both sides of the equation.

By doing this, the initially difficult task of solving for \(x\) in the exponential equation becomes a simpler problem of solving a linear equation. When we apply the natural logarithm, \(\ln\), to both sides, we turn the equation into \(\ln(7^{x+2}) = \ln(410)\). This transformation is the first crucial step in unraveling the equation and preparing it for further manipulation using the properties of logarithms.

Properties of Logarithms

The properties of logarithms are powerful tools that allow us to manipulate and simplify logarithmic expressions. One of these pivotal properties is the power rule, \( \log_b(a^n) = n \log_b(a) \), which lets us move the exponent from inside the logarithm to the front as a multiplier. In the context of our problem, after applying logarithmic transformation, we can leverage this property to rewrite the equation \( \ln(7^{x+2}) = \ln(410) \) as \( (x + 2)\ln(7) = \ln(410) \).

Another important property is the ability to isolate variables. By dividing by \( \ln(7) \) and subtracting 2, we can solve for \(x\) as shown: \( x = \frac{\ln(410)}{\ln(7)} - 2 \). Understanding these properties makes solving logarithmic equations far more approachable and demonstrates how the characteristics of logarithms facilitate the simplification of complex expressions.

Calculator Approximation

After transforming and simplifying the exponential equation using logarithmic properties, the next step to find \(x\) involves a calculator approximation. Since most logarithmic values cannot be neatly expressed as simple fractions or integers, we often turn to our trusty calculator for a decimal approximation.

In this context, we use the calculated expression \( x = \frac{\ln(410)}{\ln(7)} - 2 \) to obtain the value of \(x\) correct to two decimal places. By inputting the natural logarithms of 410 and 7 into the calculator and performing the division and subtraction, we achieve an approximate answer: \( x = 3.97 \). This approximation is not only practical but necessary to understand and interpret the solution in real-world contexts where exact answers are not always available or required.