Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log x+\log (x+3)=\log 10$$

Step-by-Step Solution

Answer

The exact solution is \(x = 2\).

Step 1: Combining The Logs On The Left

Begin by using the multiplication rule for logarithms: \(\log a + \log b = \log (ab)\). This gives: \(\log(x(x+3)) = \log(10)\)

Step 2: Convert From Logarithmic To Exponential Form

In the form \(\log_a b = c\), the logarithmic equation can be rewritten in exponential form as \(a^c = b\). Applying this, the equation becomes \(x(x+3) = 10\).

Step 3: Solving For X

Expanding out the left side gives \(x^2 + 3x - 10 = 0\). This quadratic equation can be factored into \((x - 2)(x + 5) = 0\). Setting each factor equal to zero gives the solutions \(x = 2\) and \(x = -5\).

Step 4: Check The Domain

Keep in mind that for a logarithm, the argument must be greater than 0. In the original expressions, \(x > 0\) and \(x+3 > 0\). Therefore, \(x = -5\) is not in the domain of the original logarithmic expressions and is rejected as a solution. Thus, the only valid solution is \(x = 2\).