/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Find the solution of the exponen... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 13

Find the solution of the exponential equation, correct to four decimal places. $$8^{0.4 x}=5$$

Short Answer

Expert verified
The solution is approximately \(x = 1.9342\).

Step by step solution

01

Write the Equation in Logarithmic Form

The given equation is \(8^{0.4x} = 5\). To solve for \(x\), we first need to take the logarithm of both sides. To simplify, we can use the natural logarithm (ln) or common logarithm (log). Here, we'll use the natural logarithm:\[ \ln(8^{0.4x}) = \ln(5) \]
02

Apply the Power Rule of Logarithms

The power rule of logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). Applying this rule to our equation gives:\[ 0.4x \cdot \ln(8) = \ln(5) \]
03

Solve for x

To isolate \(x\), divide both sides of the equation by \(0.4 \cdot \ln(8)\):\[ x = \frac{\ln(5)}{0.4 \cdot \ln(8)} \]
04

Calculate the Value of x

Use a calculator to compute the numerical value. First, calculate \(\ln(5)\) and \(\ln(8)\), then evaluate the expression:\[ \ln(5) \approx 1.6094 \]\[ \ln(8) \approx 2.0794 \]Substitute these into the expression:\[ x = \frac{1.6094}{0.4 \times 2.0794} \]\[ x \approx \frac{1.6094}{0.83176} \]\[ x \approx 1.9342 \]
05

Confirm the Solution is Correct to Four Decimal Places

Recompute the result to ensure accuracy if needed, double-check that all calculations match expectations. The final solution of the equation is:\[ x \approx 1.9342 \]

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Key Concepts

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

Understanding Logarithmic Form
Exponential equations like the one given can be challenging to solve directly, especially when the variable is in the exponent. That's where logarithms come into play. A logarithm answers the question: "To what power must the base be raised, to produce a given number?"

For the equation in the exercise, which is \(8^{0.4x} = 5\), converting it into logarithmic form allows us to work with the expression more easily:
  • The base of the exponent becomes the base of the logarithm.
  • The result of the exponentiation becomes the argument of the logarithm.
  • The exponent itself is what you solve for after conversion.
In this case, you use the natural logarithm \(\ln\). When applied, the equation becomes \(\ln(8^{0.4x}) = \ln(5)\). This step is crucial as it enables us to work algebraic operations on the exponents efficiently.
Applying the Power Rule of Logarithms
Once in logarithmic form, the next step is to deal with the exponent. This is where the power rule of logarithms is extremely useful. It states that \(\ln(a^b) = b \cdot \ln(a)\). Essentially, this allows us to take the exponent and multiply it with the logarithm of the base.
  • This simplifies exponential equations significantly by bringing the variable out of the exponent.
  • Once the variable is out, traditional algebraic methods can be used to isolate and solve for it.
In the exercise provided, applying this rule transforms the equation into \(0.4x \cdot \ln(8) = \ln(5)\). Now you only need to perform simple operations like division to solve for \(x\).
The Role of Natural Logarithm
The natural logarithm, often denoted as \(\ln(x)\), is a logarithm with base \(e\), where \(e\) is approximately 2.71828. It is particularly useful in calculus and in solving problems involving exponential growth and decay.
  • Natural logarithms are convenient because they can be calculated easily with modern calculators.
  • They avoid complex decimal numbers that common logarithms (base 10) might introduce.
In this example, using \(\ln\) simplifies our exponential equation significantly. By converting to natural logarithm form, the equation \(8^{0.4x}=5\) becomes manageable after applying the logarithmic rules, and you only need to compute \(\ln(8)\) and \(\ln(5)\), which are well-supported by calculators. This makes solving for \(x\) much easier, providing an exactness in results that might be more cumbersome using other logarithm bases.

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