1. We are asked to simplify the expression $7 \log_4 \left(4^{2x+5}\right) - 10 \log (x+5)$. 2. Recall the logarithm power rule: $\log_b (a^c) = c \log_b a$. Also, $\log_b b = 1$. 3. Apply the power rule to the first term: $$7 \log_4 \left(4^{2x+5}\right) = 7 (2x+5) \log_4 4 = 7 (2x+5) \times 1 = 7 (2x+5)$$ 4. So the expression becomes: $$7 (2x+5) - 10 \log (x+5)$$ 5. Distribute the 7: $$14x + 35 - 10 \log (x+5)$$ 6. This is the simplified form of the expression. --- 1. Now solve the equation $6^{x+3} - 6^x = 215$. 2. Rewrite $6^{x+3}$ as $6^x \times 6^3$: $$6^x \times 216 - 6^x = 215$$ 3. Factor out $6^x$: $$6^x (216 - 1) = 215$$ 4. Simplify inside the parentheses: $$6^x \times 215 = 215$$ 5. Divide both sides by 215: $$\cancel{215} \times 6^x = \cancel{215}$$ $$6^x = 1$$ 6. Recall that $6^0 = 1$, so: $$x = 0$$ Final answers: Simplified expression: $14x + 35 - 10 \log (x+5)$ Solution to equation: $x = 0$