Problem 363 - Precalculus

Question

Use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\)

Step-by-Step Solution

Answer

The solution to the equation is approximately \(x = 2.2405\).

Step 1: Simplify the Equation

First, move the constant on the left side of the equation to isolate the exponential term. Subtract 7.9 from both sides:\[7 e^{3x - 5} = 47 - 7.9\]which simplifies to:\[7 e^{3x - 5} = 39.1\]

Step 2: Isolate the Exponential Expression

Divide both sides by 7 to solve for the exponential expression:\[e^{3x - 5} = \frac{39.1}{7}\]which simplifies to:\[e^{3x - 5} = 5.585714285714286\]

Step 3: Apply the Natural Logarithm

Apply the natural logarithm (\(\ln\)) to both sides in order to solve for \(3x - 5\):\[\ln(e^{3x - 5}) = \ln(5.585714285714286)\]Using the logarithmic property \(\ln(e^a) = a\), this simplifies to:\[3x - 5 = \ln(5.585714285714286)\]

Step 4: Solve for \(x\)

First, calculate \(\ln(5.585714285714286)\) using a calculator, getting approximately 1.721568.Now solve for \(x\):\[3x - 5 = 1.721568\]Add 5 to both sides:\[3x = 1.721568 + 5 = 6.721568\]Divide by 3:\[x = \frac{6.721568}{3} \approx 2.240522666666667\]Rounding to the nearest ten-thousandth gives us:\[x \approx 2.2405\]

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with a base of \(e\). The number \(e\) is a mathematical constant approximately equal to 2.71828. Natural logarithms are widely used in mathematics, especially when dealing with exponential growth or decay.
The special property of natural logarithms is that they are the inverse operations of exponential functions. If you apply the natural logarithm to an exponential function where the base is \(e\), you're essentially "undoing" the exponential part. For example:

• \(\ln(e^a) = a\)

This property is very useful for solving exponential equations. In the given equation \(7 e^{3x-5} + 7.9 = 47\), we isolated the exponential term to apply \(\ln\) and solve for \(x\).
Using a calculator, we can find \(\ln(5.5857)\) which helps us solve the equation effectively. This ability to "open up" exponentials with natural logs is essential in math and various scientific fields.

Exponential Functions

Exponential functions are equations where the variable appears in the exponent. These functions often model real-world situations such as population growth, radioactive decay, and interest calculations. The most common exponential function is of the form \(a e^{bx}\), where \(e\) is the base of the natural logarithm.
Exponential functions grow very quickly, and they are used in equations like the one you solved, \(7 e^{3x - 5} = 5.5857\). This type of function is characterized by repeated multiplication, in this case involving powers of \(e\).
When solving exponential equations:

• First, try to isolate the exponential expression.
• Once isolated, apply logarithms to simplify.

This approach allows us to handle the equation non-linearly and find solutions in terms of the variable \(x\).
Understanding exponential functions is critical as they appear frequently in many areas of science and engineering.

Rounding Numbers

Rounding numbers is a mathematical process by which we approximate a number to make it easier to work with.
In many cases, especially when dealing with decimal numbers resulting from calculations, it's practical to round.To round to the nearest ten-thousandth:

• Identify the digit in the ten-thousandth place.
• Look at the next digit (to the right) to determine whether to round up or stay the same.
  For example, when we calculate \(x \approx 2.24052266\), we look at the digit after the ten-thousandths place (\(2\) in this case).
• If it's 5 or greater, round up; otherwise, stay the same.

In our example, \(x\) becomes approximately \(2.2405\). This method ensures our results are manageable and meet specified requirements. Rounding is important, especially in science, finance, and engineering, where exact values often need to be expressed within a certain tolerance.