8. Step 1: Express both sides of the equation with the same base. The equation is $25^{3x-5} = 5$. Since $25 = 5^2$, we can rewrite the equation as: $$ (5^2)^{3x-5} = 5^1 $$ Step 2: Apply the power rule for exponents, $(a^m)^n = a^{mn}$. $$ 5^{2(3x-5)} = 5^1 $$ $$ 5^{6x-10} = 5^1 $$ Step 3: Equate the exponents, as the bases are now the same. $$ 6x-10 = 1 $$ Step 4: Solve for $x$. $$ 6x = 1 + 10 $$ $$ 6x = 11 $$ $$ x = \frac{11}{6} $$ The value of $x$ is: $$ \boxed{\frac{11}{6}} $$ 9. Step 1: Rewrite the expression using reciprocals. The expression is $\frac{5}{0.6699} - \frac{1}{4.231}$. This can be written as $5 \times \frac{1}{0.6699} - \frac{1}{4.231}$. Step 2: Find the reciprocal values. Using a calculator to simulate reciprocal table values: $$ \frac{1}{0.6699} \approx 1.4928 $$ $$ \frac{1}{4.231} \approx 0.2364 $$ Step 3: Substitute the reciprocal values into the expression and calculate. $$ 5 \times 1.4928 - 0.2364 $$ $$ 7.4640 - 0.2364 $$ $$ 7.2276 $$ The evaluated expression is: $$ \boxed{7.2276} $$ 10. Step 1: Evaluate the logarithm. The given equation is $\log 0.00001 = 2y$. Assuming the base of the logarithm is 10 (common for "log" when no base is specified). First, express $0.00001$ as a power of 10: $$ 0.00001 = 10^{-5} $$ Now, evaluate $\log_{10} (10^{-5})$: $$ \log_{10} (10^{-5}) = -5 $$ Step 2: Solve for $y$. Substitute the value of the logarithm back into the given equation: $$ -5 = 2y $$ Divide by 2 to find $y$: $$ y = \frac{-5}{2} $$ $$ y = -2.5 $$ The value of $y$ is: $$ \boxed{-2.5} $$ Send me the next one 📸