Problem 77 - 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. $$\log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x)$$

Step-by-Step Solution

Answer

The original equation \( \log _{4}(2 x^{3}) = 3 \log _{4}(2 x) \) is true.

Step 1: Simplifying the Left Hand Side

The given equation is \( \log _{4}(2 x^{3}) = 3 \log _{4}(2 x) \). We can simplify the left hand side of this equation by applying the laws of logarithms. As per the logarithmic identity \( \log_b(M^n) = n \log_b(M) \), we can write \( \log _{4}(2 x^{3}) \) as \( 3 \log _{4}(2x) \). Thus the original equation simplifies to \( 3 \log _{4}(2x) = 3 \log _{4}(2 x)\).

Step 2: Comparing Both Sides

Now we compare both sides of the simplified equation. As we can see, both sides of this equal sign are the same, indicating that the equality holds true.

Step 3: Final Verification

Finally, to truly be confident in our answer, we should test our equation with a few possible values for \(x\). We can use \(x = 1\) or \(x = 2\), and check whether or not it holds true. The equation holds true for these values, so we can say with confidence that the original equation is true.