Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln (5 x)+\ln 1=\ln (5 x)$$

Step-by-Step Solution

Answer

The original equation \(\ln (5x) + \ln 1 = \ln (5x)\) is true.

Step 1: Analyze the Equation

The equation is given as \(\ln (5x) + \ln 1 = \ln (5x)\). The base of the logarithm is natural (e). Let's proceed and apply the logarithm laws to this equation.

Step 2: Apply the Logarithm Law

According to the product rule of logarithms, the sum of two logs with the same base equals the log of the product of the quantities. Therefore, the left side of the equation can be written as \(\ln (5x \cdot 1)\).

Step 3: Simplify the Equation

The product \(5x \cdot 1\) simplifies to \(5x\), then our equation becomes \(\ln (5x) = \ln (5x)\).

Step 4: Check the Solution

Now, one thing to note is that the function \(\ln x\) is a one-to-one function, meaning that if \(\ln a = \ln b\), then \(a = b\). It follows that if \(\ln (5x) = \ln (5x)\), then \(5x = 5x\), which is a true statement. Therefore, the original equation \(\ln (5x) + \ln 1 = \ln (5x)\) is also true.

Step 5: Conclusion

As a result of our performed steps, no changes needed to be made to the original equation as it is already true. The initial expression provided was correct under the established logarithm rules.