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Chapter 7: Problem 174

$$ x^{3 \log x-\frac{1}{\log x}}=\sqrt[3]{10} $$

Short Answer

Expert verified
The solution to the given equation \(x^{3 \log x-\frac{1}{\log x}}=\sqrt[3]{10}\) can be found by rewriting the equation in exponential form, applying logarithm properties, and solving for x. By using numerical methods, such as the Newton-Raphson Method, we find that the approximate value of x is \(x \approx 1.4356\).

Step by step solution

01

Rewrite the equation in exponential form

To rewrite the given equation, we need to express the right-hand side in exponential form. Since the cube root is the same as raising to the power of \(\frac{1}{3}\), we have: \(x^{3 \log x-\frac{1}{\log x}}=10^{\frac{1}{3}}\)
02

Apply logarithm to both sides of the equation

To remove the exponent from the left-hand side, take the natural logarithm of both sides: \(\log(x^{3 \log x-\frac{1}{\log x}})=\log\left(10^{\frac{1}{3}}\right)\)
03

Use logarithm properties to simplify the equation

Using the property \(\log(a^b)=b\log a\), we can simplify the left side of the equation: \((3 \log x-\frac{1}{\log x})\log{x}=\frac{1}{3}\log{10}\)
04

Isolate x terms on the left side

First, replace the \(\log{x}\) terms with a variable, say \(y\). The equation becomes: \(3y^2 - \frac{1}{y}= \frac{1}{3}\log{10}\) Now, multiply both sides by \(3y\) to remove the fraction: \(3y^3 - 3 = y\log{10}\)
05

Solve for y

We see this is a difficult non-linear equation to solve analytically. One option would be to numerically approximate the value of y. By using numerical methods, such as the Newton-Raphson Method or a graphing calculator, we can approximate the value of y: \(y \approx 0.3614\)
06

Solve for x

Now that we have an approximation for y, we need to find the corresponding value of x. Since \(y = \log{x}\), we can rewrite this as: \(x = e^y\) Plugging in the approximate value of y: \(x \approx e^{0.3614}\) \(x \approx 1.4356\) Therefore, the approximate solution to the given equation is \(x \approx 1.4356\).

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Key Concepts

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

Logarithmic Properties
Understanding logarithmic properties is crucial when working with exponential equations. Logarithms, the inverse functions of exponentials, transform multiplication and division into addition and subtraction, making complex calculations more manageable.

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