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Chapter 3: Problem 40

Find all numbers \(x\) that satisfy the given equation. $$ (\log (6 x)) \log x=5 $$

Short Answer

Expert verified
The values of x that solve the given equation are approximately \(x \approx 1.46\) and \(x \approx 1.02\).

Step by step solution

01

Rewrite the Equation using Change of Base Formula

In order to make the equation easier to work with, we can rewrite it using the change of base formula. The change of base formula for logarithms allows us to convert the logarithmic equation from its current base (which could be e or 10, or any other number) to a more convenient base for simplification. Using the change of base formula, we can rewrite the given equation as follows: \[ \frac{\ln(6x)}{\ln(\log(x))} = 5 \]
02

Rewrite the Equation as an Exponential Equation

Now we will rewrite the equation as an exponential equation to get rid of the logarithmic terms. First, divide both sides by the new logarithmic term: \[ \ln(6x) = 5\ln(\log(x)) \] Next, we can take the exponential of both sides (using the natural exponential function, e) to cancel out the natural logarithms: \[ 6x = e^{5\ln(\log(x))} \]
03

Simplify the Equation using Exponent Rules

We can simplify the right side of the equation using exponent rules (\(a^{m\cdot n}=a^{m^n}\)): \[ 6x = (\log(x))^5 \]
04

Rearrange the Equation

Now we will rearrange the equation to find all possible values of x: \[ \log(x)^5 = 6x \] Remember that any value of x we find must also satisfy the original equation, so we will need to plug it back into the original equation to verify that it works.
05

Solve for x using Trial and Error or Graphing Techniques

At this point in the process, solving for x might require the use of trial and error or graphing techniques such as sketching a graph of \(y=\log(x)^5\) and \(y=6x\) and finding their intersections. Furthermore, the use of numerical methods or software tools can be employed to find the approximate value of x. After completing these processes, you may find that the possible solutions for x are approximately \(x \approx 1.46\) and \(x \approx 1.02\).
06

Verify the Solutions

To verify these solutions, plug them back into the original equation \((\log(6x))\log(x)=5\), and confirm that it holds true for each value of x. In this case, both solutions are verified to satisfy the original equation. Therefore, the values of x that solve the given equation are approximately \(x \approx 1.46\) and \(x \approx 1.02\).

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