Homework 10 (not due for credit)

---

Directions:

1. Show each step of your work and fully simplify each expression.
2. Turn in your answers in class on a physical piece of paper.
3. Staple multiple sheets together.
4. Feel free to use Desmos for graphing.

---

Answer the following:

1. State which of the following are true or false:
   1. $\log_3 4^5 = 5$
      Hint Different bases.
   2. $\ln 3x = \ln 3 + \ln x$
   3. $\ln(3 + x) = \ln 3 + \ln x$
   4. $\ln(x(x-1)) = \ln x + \ln (x - 1)$
   5. $\ln x^2 = 2\ln x$
   6. $(\ln x)^2 = 2\ln x$
   7. $\log_2(x - 3) = \log_2 x - \log_2 3$
   8. $\ln\left(\dfrac{x}{x - 1}\right) = \ln x - \ln (x - 1)$
2. Use Laws of Logarithms to evaluate the expression without a calculator:
   1. $\log 5 + \log 20$
   2. $\log_5 \sqrt{5}$
   3. $\log_2 60 - \log_2 15$
   4. $\log_4 16^{100}$
   5. $\ln (\ln e^{e^{200}})$
3. Expand these logarithms using Laws of Logarithms:
   1. $\log_3 4x^2$
      Hint $4$ and $x^2$ are factors
   2. $\ln \left(x^2 (x-1)\right)^3$
   3. $\ln \dfrac{3x^2}{(x - 1)^4}$
   4. $\ln \sqrt[3]{\dfrac{a}{b + 2}}$
   5. $\log_7 ab^2$
   6. $\ln(\ln(x)\ln(y))$
4. Combine these logarithms using Laws of Logarithms:
   1. $\log_4 x + \log_4 (x - 1)$
   2. $3\ln (x-3) - 2\ln x$
   3. $\frac{1}{3}\log (2x^2 + 1) - \frac{1}{2}\log(3x - 4)$
5. Find the solution of the exponential equation. Write your answer as a number unless the logarithm is not easy to evaluate.
   1. $5^{x - 1} = 125$
   2. $5^{2x-3} = 1$
   3. $e^{1-4x} = e^2$
   4. $5^x = 4^{x+1}$
      Hint Both sides have different bases. First convert $4^{x+1} = 4\cdot 4^x$ and divide both sides by $4^x$.
6. Find the solution of the logarithmic equation. Write your answer as a number unless the exponent is not easy to evaluate.
   1. $\log x + \log(x - 1) = \log(4x)$
   2. $\ln x = 10$
   3. $\ln(2 + x) = 1$
   4. $2\log x = \log 2 + \log(3x - 4)$
   5. $\log x + \log(x - 3) = 1$
   6. $\log_2(x^2 - x - 2) = 2$
   7. $\log_2(\log_3 x) = 2$
      Hint Convert to exponential form twice.