Problem 147 - College Algebra Essentials

Question

Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Step-by-Step Solution

Answer

The solutions are \( x = 100 \) and \( x = 0.001 \).

Step 1: Simplify the equation

Use the properties of logarithms to simplify the given equation. The property \( \log_a(uv) = \log_a(u) + \log_a(v) \) is applied here, which turns equation \( (\log x)(2 \log x+1)=6 \) into \( \log^2x + \log x - 6 = 0 \). This is a quadratic equation.

Step 2: Factor the quadratic equation

The equation is factored into two binomial expressions as \( (\log x - 2)(\log x + 3) = 0 \). This is done by seeking two numbers which multiply to -6 and add to 1. For such, -2 and 3 are found.

Step 3: Solving for x

Setting each binomial expression equals to 0 then find the values of x. If \( \log x - 2 = 0 \), then \( \log x = 2 \) and \( x = 10^2 = 100 \). If \( \log x + 3 = 0 \), then \( \log x = -3 \) and \( x = 10^{-3} = 0.001 \).

Step 4: Checking solutions

Substitute the x-values obtained in step 3 into the original equation \( (\log x)(2 \log x+1)=6 \) for verification. It's found out that both values satisfy the equation.