To show the equality given in the equation:

$\log_{b^4} x^2 = \log_b \sqrt{x}$

we can start by rewriting both sides using logarithmic properties and simplifying step-by-step.

Step 1: Rewrite the Left Side

The left side is $\log_{b^4} x^2$. Using the change of base formula for logarithms, we can rewrite it as follows:

$\log_{b^4} x^2 = \frac{\log_b x^2}{\log_b b^4}$

Since $\log_b x^2 = 2 \log_b x$ by the power rule, and $\log_b b^4 = 4$ because $b^4$ is $b$ raised to the power 4, we get:

$\log_{b^4} x^2 = \frac{2 \log_b x}{4} = \frac{1}{2} \log_b x$

Step 2: Rewrite the Right Side

The right side of the equation is $\log_b \sqrt{x}$. We can simplify this using the power rule for logarithms:

$\log_b \sqrt{x} = \log_b x^{1/2} = \frac{1}{2} \log_b x$

Step 3: Compare Both Sides

Now we have both sides of the equation simplified as follows:

$\frac{1}{2} \log_b x = \frac{1}{2} \log_b x$

Since both sides are equal, we have shown that:

$\log_{b^4} x^2 = \log_b \sqrt{x}$

Therefore, the given equation holds true.

Do you want further clarification on any step or more details?

Related Questions

1. How does the change of base formula work for logarithms in general?
2. What are the properties of logarithms that simplify expressions involving exponents?
3. How do we derive the power rule for logarithms?
4. What other methods could be used to simplify logarithmic expressions like $\log_{b^4} x^2$?
5. Why does the base $b$ not affect the equality in this case?

Tip: Always remember to apply the power rule, $\log_b(x^k) = k \log_b(x)$, when you encounter an exponent inside a logarithmic function. This rule can simplify logarithmic expressions significantly.