1. The problem involves matching algebraic expressions and logarithmic equations to their corresponding graphs and solving exponential and logarithmic equations. 2. For the algebraic expressions and graphs: - $y = x^2 + 6$ is a parabola shifted up by 6. - $y = |x - 5| - 1$ is an absolute value function shifted right by 5 and down by 1. - $y = \sqrt{x + 2}$ is a square root function shifted left by 2. - $y = |x| - 8$ is an absolute value function shifted down by 8. - $y = -(x - 5)^2$ is a downward parabola shifted right by 5. 3. For the exponential equations: - Solve $2^x = 32$: Since $32 = 2^5$, then $x = 5$. - Find the intercept of $y = 4^x - 1$ at $x = 5$: Calculate $y = 4^5 - 1 = 1024 - 1 = 1023$. - Solve $9^{x+2} = \frac{1}{27}$: Rewrite bases: $9 = 3^2$, $\frac{1}{27} = 3^{-3}$. So, $(3^2)^{x+2} = 3^{-3}$ implies $3^{2(x+2)} = 3^{-3}$. Equate exponents: $2(x+2) = -3$. Solve: $2x + 4 = -3 \Rightarrow 2x = -7 \Rightarrow x = -\frac{7}{2}$. - Solve $4^{2x} = 16$: Rewrite $16 = 4^2$. So, $4^{2x} = 4^2$ implies $2x = 2$. Solve: $x = 1$. - Solve $7^{x-1} + 7 = 8$: Subtract 7: $7^{x-1} = 1$. Since $7^0 = 1$, then $x - 1 = 0$. Solve: $x = 1$. - Solve $25^{3x} = 5^{x+1}$: Rewrite $25 = 5^2$. So, $(5^2)^{3x} = 5^{x+1}$ implies $5^{6x} = 5^{x+1}$. Equate exponents: $6x = x + 1$. Solve: $5x = 1 \Rightarrow x = \frac{1}{5}$. - Solve $5^{2x+1} = 125$: Rewrite $125 = 5^3$. So, $5^{2x+1} = 5^3$ implies $2x + 1 = 3$. Solve: $2x = 2 \Rightarrow x = 1$. - Solve $\left(\frac{1}{2}\right)^x = \left(\frac{1}{2}\right)^7$: Equate exponents: $x = 7$. 4. For the logarithmic expressions and flowchart: - $\log_7 1 = 0$ because any log base of 1 is 0. - $\log_3 \frac{1}{9} = \log_3 3^{-2} = -2$. - $\log_2 16 = 4$ because $2^4 = 16$. - $\log_9 9 = 1$. - $\log_{16} 4 = \frac{1}{2}$ because $16^{1/2} = 4$. - $3^{\log_3 5} = 5$ by the property of logs and exponents. - $\log_9 g^{21} = 21 \log_9 g$. - $\log_5 25 = 2$ because $5^2 = 25$. 5. The path to the trophy involves evaluating these expressions and following the edges with correct values. Final answers: - $x$ for $2^x=32$ is $5$. - Intercept of $y=4^x-1$ at $x=5$ is $1023$. - $x$ for $9^{x+2} = \frac{1}{27}$ is $-\frac{7}{2}$. - $x$ for $4^{2x} = 16$ is $1$. - $x$ for $7^{x-1} + 7 = 8$ is $1$. - $x$ for $25^{3x} = 5^{x+1}$ is $\frac{1}{5}$. - $x$ for $5^{2x+1} = 125$ is $1$. - $x$ for $\left(\frac{1}{2}\right)^x = \left(\frac{1}{2}\right)^7$ is $7$. - Logarithmic values as above.