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Chapter 3: Problem 95

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. $$\ln (5 x)+\ln 1=\ln (5 x)$$

Short Answer

Expert verified
The given equation \( \ln (5x) + \ln 1 = \ln (5x) \) is true for all \( x > 0 \) without any changes needed.

Step by step solution

01

Start with the original equation

The given equation is \( \ln (5x) + \ln 1 = \ln (5x) \). This equation suggests that \( \ln (5x) + \ln 1 \) is equal to \( \ln (5x) \). It's important to remember that \( \ln 1 = 0 \). So, we can simplify.
02

Substitute \(\ln 1 = 0\) into the equation

By substituting \( \ln 1 = 0 \) into the original equation, the equation becomes \( \ln (5x) + 0 = \ln (5x) \). This equation simplifies to \( \ln (5x) = \ln (5x) \), which is an identity because it is true for all values of x (under the condition that \( x > 0 \) because \( ln(a) \) is defined when \( a > 0 \))
03

Verify the given equation is true

Since \( \ln (5x) = \ln (5x) \) is true for all applicable values of x, the original equation \( \ln (5x) + \ln 1 = \ln (5x) \) is also true for these values. So, no changes are needed to be made to the equation for it to be true.

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