Problem 113 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2$$

Step-by-Step Solution

Answer

The solution to the equation would be rounded to three decimal places. Finally, the verification of these solutions must be done graphically.

Step 1: Combine the logarithmic terms

The formula used here is \(\log_ba - \log_bc = \log_b (a/c)\). Applying it to the given equation we get : \[\log_{10} \left(\frac{8x}{1+\sqrt{x}}\right) = 2 \]

Step 2: Convert the logarithm equation into an exponential equation

The conversion formula is \(b^y = a\). Applying it to the updated equation gives us:\[\left(\frac{8x}{1+\sqrt{x}}\right) = 10^2 \]

Step 3: Simplify the equation

Solving the exponential equation:\[8x = 100*(1+\sqrt{x}) \]will result in a quadratic equation.

Step 4: Solve the Quadratic equation

First, the equation should be rearranged to the form \(ax^2 + bx + c = 0\). Then, use the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)]/(2a)\) to find the solutions, with \(a\), \(b\), and \(c\) being the coefficients of \(x^2\), \(x\), and the constant term respectively.

Step 5: Check the solutions

Finally, solutions have to be checked by substituting them into the original equation. List out those solutions which satisfy the equation. Important to note, negative solutions and zero should not be considered as they are out of domain for the logarithmic functions.

Step 6: Round the result

Round off the final solutions to three decimal places.

Step 7: Verify the solution using a graphing utility

To verify the solutions, plot a graph of the function provided in the equation and see if the solutions lie on the curve.

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is essential when solving equations involving logarithms. A logarithm, generally expressed as \( \log_b a \), is the power to which the base \( b \) must be raised to obtain the number \( a \) . The most commonly used properties are: