91Ó°ÊÓ

(a) To solve the equation \(10^{2 x}=93\), we need to take logarithm of both sides to the base of 10. Applying the logarithm property \(\log a^{b} = b \log a\), we get \(2x \log 10 = \log 93\), which simplifies to \(x = \frac{\log 93}{2 \log 10}\). (b) For \(10^{3 x+2}=1,000,000\), we proceed similarly and get \(3x + 2 = \log 1,000,000 \Rightarrow x = \frac{\log 1,000,000 - 2}{3}\). (c) For \(2^{x+1}=7\), we get \(x = \frac{\log 7 - \log 2}{\log 2}\). (d) \(3^{x} 3^{x^{2}}=3\) simplifies to \(x + x^2 = 1 \Rightarrow x^2 + x - 1 = 0\), which can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). (e) \(5 B^{x}=(2 C)^{x+1}\) can be written as \(B^{x} = \frac{(2C)^{x+1}}{5}\), which can be solved similarly.

(f) For \(\ln x+2=5\), subtract 2 from both sides and take the exponent of both sides to get \(x = e^{5-2}\). (g) To find \(x\) from \(\log _{10} x=17\), raise both sides as powers of 10 to get \(x = 10^{17}\). (h) For \(\ln (5 x-40)=3\), raise both sides as powers of e to get \(5x-40 = e^{3}\) and then solve for \(x\). (i) For \(\log _{10}\left(2 x^{2}+4\right)=2\), raise both sides as powers of 10 to get \(2x^{2}+4 = 10^{2}\), which simplifies to a quadratic equation. (j) \(3 \cdot 2^{x / 7}-4=12\) can be simplified as \(2^{x / 7} = \frac{12+4}{3}\) and take natural logarithm on both sides.