Here's the solution to your problems: 6. Solve for $x$ in $2^{2x} - 18 \times 2^x = 40$ Step 1: Rewrite the equation using exponent rules. We know that $2^{2x} = (2^x)^2$. So the equation becomes: $$(2^x)^2 - 18 \times 2^x = 40$$ Step 2: Introduce a substitution to form a quadratic equation. Let $y = 2^x$. Substitute $y$ into the equation: $$y^2 - 18y = 40$$ Step 3: Rearrange the equation into standard quadratic form and solve for $y$. $$y^2 - 18y - 40 = 0$$ Factor the quadratic equation: $$(y - 20)(y + 2) = 0$$ This gives two possible values for $y$: $$y - 20 = 0 \implies y = 20$$ $$y + 2 = 0 \implies y = -2$$ Step 4: Substitute back $2^x$ for $y$ and solve for $x$. Case 1: $y = 20$ $$2^x = 20$$ Take the logarithm of both sides (e.g., base 10 or natural log): $$\log(2^x) = \log(20)$$ $$x \log(2) = \log(20)$$ $$x = \frac{\log(20)}{\log(2)}$$ Using a calculator: $$x \approx \frac{1.30103}{0.30103} \approx 4.3219$$ Case 2: $y = -2$ $$2^x = -2$$ An exponential function with a positive base, like $2^x$, is always positive for real values of $x$. Therefore, there is no real solution for $x$ in this case. The only real solution for $x$ is: $$\boxed{x = \frac{\log(20)}{\log(2)} \approx 4.3219}$$ 7. Given that $\text{Sin}(x - 30)^0 - \text{Cos}(4x)^0 = 0$. Find the $\text{tan}(2x+30)^0$ Step 1: Rearrange the given equation. $$\text{Sin}(x - 30)^0 = \text{Cos}(4x)^0$$ Step 2: Use the trigonometric identity $\text{Sin}(\theta) = \text{Cos}(90^0 - \theta)$. This means that the angles must be complementary (or differ by multiples of $360^0$, but for basic problems, we assume the simplest case). So, we can equate the angles: $$(x - 30)^0 + (4x)^0 = 90^0$$ Step 3: Solve for $x$. $$x - 30 + 4x = 90$$ $$5x - 30 = 90$$ $$5x = 90 + 30$$ $$5x = 120$$ $$x = \frac{120}{5}$$ $$x = 24$$ Step 4: Substitute the value of $x$ into the expression $\text{tan}(2x+30)^0$. First, calculate the angle: $$2x + 30 = 2(24) + 30$$ $$2x + 30 = 48 + 30$$ $$2x + 30 = 78$$ Now, find the tangent of this angle: $$\text{tan}(78)^0$$ Using a calculator: $$\text{tan}(78)^0 \approx 4.7046$$ The value is: $$\boxed{\text{tan}(2x+30)^0 \approx 4.7046}$$ 3 done, 2 left today. You're making progress.