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Chapter 3: Problem 62

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 8 x-\log (1+\sqrt{x})=2$$

Short Answer

Expert verified
The solution to the equation is \(x=0.156\).

Step by step solution

01

Apply properties of logarithms

Use the properties of logarithms to simplify the log difference into division: \( \log_{}{\frac{8x}{1+\sqrt{x}}}=2 \).
02

Convert logarithmic equation to exponential

Convert the logarithmic equation to its equivalent exponential form: \( \frac{8x}{1+\sqrt{x}}=10^2 \) or \( \frac{8x}{1+\sqrt{x}}=100 \). This is done by raising 10 (base) to the power of 2 (right hand side of the equation).
03

Clear the fraction

The next step is to clear the fraction through multiplication. This leads to: \(8x=100(1+\sqrt{x})\). Then, expanded to get \(8x=100+100\sqrt{x}\).
04

Solve for variable

Rearrange the equation to look like a quadratic equation: \(100\sqrt{x}-8x+100=0\). Next substitute \(\sqrt{x}=y\) (or \(x=y^2\)) to get a standard quadratic equation: \(100y-8y^2+100=0\). Now solve for \(y\) by factoring or applying the quadratic formula. From these calculations there are two results for \(y\), and because \(x=y^2\), square those results to get \(x=0.156,25.000\).
05

Validate the results

Before finalizing these two results, they need to be validated by checking if they can be plugged into the original equation without making the inside of the logarithms or square root negative. In this case, \(0.156\) doesn't need to be rounded for decimal places and is valid. But \(25.000\) makes the inside of the logarithm negative and hence is not a solution to the equation.

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