1. The problem is to evaluate $\log_{90} 3 \times \log_{10} 3$. 2. Recall the change of base formula: $\log_a b = \frac{\log_c b}{\log_c a}$ for any positive $c \neq 1$. 3. Using natural logarithms (ln) for convenience, rewrite each logarithm: $$\log_{90} 3 = \frac{\ln 3}{\ln 90}$$ $$\log_{10} 3 = \frac{\ln 3}{\ln 10}$$ 4. Multiply the two expressions: $$\log_{90} 3 \times \log_{10} 3 = \frac{\ln 3}{\ln 90} \times \frac{\ln 3}{\ln 10} = \frac{(\ln 3)^2}{\ln 90 \times \ln 10}$$ 5. Factorize $90$ to simplify $\ln 90$: $$90 = 9 \times 10$$ $$\ln 90 = \ln (9 \times 10) = \ln 9 + \ln 10$$ 6. Substitute back: $$\frac{(\ln 3)^2}{(\ln 9 + \ln 10) \times \ln 10}$$ 7. Note that $\ln 9 = \ln (3^2) = 2 \ln 3$. 8. So the denominator becomes: $$(2 \ln 3 + \ln 10) \times \ln 10$$ 9. The expression is now: $$\frac{(\ln 3)^2}{(2 \ln 3 + \ln 10) \ln 10}$$ 10. This is the simplified exact form. For a numerical approximation: - $\ln 3 \approx 1.0986$ - $\ln 10 \approx 2.3026$ Calculate denominator: $$2 \times 1.0986 + 2.3026 = 2.1972 + 2.3026 = 4.4998$$ Multiply by $\ln 10$: $$4.4998 \times 2.3026 \approx 10.36$$ Numerator: $$(1.0986)^2 \approx 1.2069$$ Final value: $$\frac{1.2069}{10.36} \approx 0.1165$$ Final answer: $\log_{90} 3 \times \log_{10} 3 \approx 0.1165$