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Chapter 4: Problem 36

Solve the following equations for \(x.\) $$e^{5 x} \cdot e^{\ln 5}=2$$

Short Answer

Expert verified
x = \frac{\ln 2 - \ln 5}{5}

Step by step solution

01

Simplify the expression

Combine the exponents using the property of exponents: \( e^a \times e^b = e^{a + b} \). Here, this results in: \[ e^{5x} \times e^{\ln 5} = e^{5x + \ln 5} \].
02

Set the simplified version equal to 2

The equation now is: \[ e^{5x + \ln 5} = 2 \].
03

Apply natural logarithm to both sides

Take the natural logarithm (\(\ln\)) of both sides to remove the exponential function: \[ \ln(e^{5x + \ln 5}) = \ln(2) \].
04

Use the property of logarithms

Since \(\ln(e^a) = a\), the equation simplifies to: \[ 5x + \ln 5 = \ln 2 \].
05

Solve for x

To isolate \(x\), rearrange the equation: \[ 5x = \ln 2 - \ln 5 \] and then divide by 5: \[ x = \frac{\ln 2 - \ln 5}{5} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Exponents
Exponential equations often require the use of properties of exponents to solve them. In this exercise, we use the property: \( e^a \cdot e^b = e^{a + b} \). This helps us combine the exponents into a single expression.
Remember, you can add the exponents when you are multiplying like bases. This makes solving exponential equations more manageable.
Some key properties of exponents to keep in mind are:
  • \(a^m \cdot a^n = a^{m+n}\)
  • \(\left(a^m\right)^n = a^{m \cdot n}\)
  • \(\frac{a^m}{a^n} = a^{m-n}\)
These properties allow us to manipulate and simplify exponential expressions.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is the inverse function of the exponential function with base \(e\). This relationship helps us solve equations involving \(e\) by 'undoing' the exponential operation.
For example, taking the natural logarithm of \(e^x\) yields: \(\ln(e^x) = x\).
In our exercise, applying the natural logarithm to both sides of the equation \(e^{5x + \ln 5} = 2\) allows us to simplify it to a linear form, making it easier to solve: \(\ln(e^{5x + \ln 5}) = \ln(2)\). This simplifies to: \(5x + \ln 5 = \ln 2\)
Understanding the natural logarithm is key to solving many exponential equations effectively.
Solving for x
After simplifying an exponential equation or using logarithms, the final step is solving for the variable \(x\). In our problem, we arrive at the equation: \(5x + \ln 5 = \ln 2\).
To isolate \(x\), we follow these steps:
  • Subtract \(\ln 5\) from both sides: \(5x = \ln 2 - \ln 5\)
  • Divide by 5: \(x = \frac{\ln 2 - \ln 5}{5}\)
Breaking down each step ensures you maintain clarity and avoid mistakes.
This process shows that solving exponential equations often requires combining properties of exponents with logarithmic functions. Always double-check each step to ensure accuracy.

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