1. **Problem 1:** Solve the equation $|x+1| = e$. - The absolute value equation $|x+1| = e$ means $x+1 = e$ or $x+1 = -e$. - Since $e$ is positive, both cases are valid. 2. **Solution for Problem 1:** - Case 1: $x+1 = e \implies x = e - 1$ - Case 2: $x+1 = -e \implies x = -e - 1$ --- 3. **Problem 2:** Solve the equation $\frac{z}{x+1} = (e^3)^{x+1}$. - Rewrite the right side using exponent rules: $(e^3)^{x+1} = e^{3(x+1)}$. 4. **Solution for Problem 2:** - The equation becomes $\frac{z}{x+1} = e^{3(x+1)}$. - Multiply both sides by $x+1$ (assuming $x \neq -1$): $$z = (x+1) e^{3(x+1)}$$ - This expresses $z$ in terms of $x$. --- 5. **Problem 3:** Solve the logarithmic equation $\ln(x + 70) + \ln x = \ln 7$. - Use the logarithm property: $\ln a + \ln b = \ln(ab)$. 6. **Solution for Problem 3:** - Combine logs: $\ln((x+70) x) = \ln 7$ - Since $\ln A = \ln B \implies A = B$, we get: $$x(x+70) = 7$$ - Expand: $$x^2 + 70x = 7$$ - Rearrange: $$x^2 + 70x - 7 = 0$$ 7. **Solve quadratic equation:** - Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=70$, $c=-7$. - Calculate discriminant: $$\Delta = 70^2 - 4 \times 1 \times (-7) = 4900 + 28 = 4928$$ - Calculate roots: $$x = \frac{-70 \pm \sqrt{4928}}{2}$$ - Approximate $\sqrt{4928} \approx 70.21$. - Roots: $$x_1 = \frac{-70 + 70.21}{2} \approx 0.105$$ $$x_2 = \frac{-70 - 70.21}{2} \approx -70.105$$ 8. **Check domain:** - Since $\ln x$ requires $x > 0$, discard $x_2$. - Final solution: $x \approx 0.105$. --- **Final answers:** - Problem 1: $x = e - 1$ or $x = -e - 1$ - Problem 2: $z = (x+1) e^{3(x+1)}$ - Problem 3: $x \approx 0.105$