Problem 78 - 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. $$\ln \left(8 x^{3}\right)=3 \ln (2 x)$$

Step-by-Step Solution

Answer

The given equation \(\ln(8x^3) = 3 \ln(2x)\) is True. It does not need any changes.

Step 1: Apply logarithmic power rule

Apply the power rule of logarithm on the left side of the equation: \(\ln(8x^3) = \ln(8) + \ln(x^3)\). Apply the same power rule on the right side of the equation: \(3 \ln (2x) = \ln {(2x)}^3\)

Step 2: Further breakdown of logarithmic expressions

Breaking down further, from step 1 \(\ln(8) + \ln(x^3) = \ln(2^3) + 3 \ln(x)\). On the other side \(\ln {(2x)}^3 = \ln(8x^3)\). So, both sides are equal.

Step 3: Verify the Equality

The \(2^{nd}\) step verifies that \(\ln(8x^3) = \ln(8x^3)\), so the given statement is already true, hence no changes are required to make it true.