1. The problem involves understanding logarithmic properties and evaluating expressions with exponents. 2. The logarithmic identities given are: - $\log(ab) = \log a + \log b$ - $\log \frac{a}{b} = \log a - \log b$ 3. These identities are fundamental rules of logarithms that help simplify expressions involving products and quotients inside a logarithm. 4. Next, consider the expression $\frac{x^{2^2}}{n+5}$. - First, evaluate the exponent $2^2 = 4$. - So, the numerator becomes $x^4$. - The expression simplifies to $\frac{x^4}{n+5}$. 5. The equation $z^2 = 2^{n+5}$ relates $z$ and $n$. - To solve for $z$, take the square root of both sides: $$z = \sqrt{2^{n+5}} = 2^{\frac{n+5}{2}}$$ 6. This shows how exponents and roots interact: the square root of an exponential expression is the exponential raised to half the power. 7. Summary: - Logarithm product and quotient rules help simplify log expressions. - $\frac{x^{2^2}}{n+5} = \frac{x^4}{n+5}$. - $z = 2^{\frac{n+5}{2}}$ from $z^2 = 2^{n+5}$. These steps clarify the use of logarithmic identities and exponent rules in algebraic expressions.