1. Write down an explanation of how you can estimate the value of $\log_{10}(31415926)$ without appealing to technology.
2. Without appealing to technology, figure out the exact value of the following logarithms. (Hint: they’re all whole numbers.)

   $\log_5(625)$

   $\log_2(1024)$

   $\log_{11}(14641)$

   $\ln\bigl(\mathrm{e}^7\bigr)$

3. Using a calculator, write down a decimal approximation of each of the following numbers rounded to two decimal places.

   $\log_{10}(31415926)$

   $\ln(100)$

   $\log_5(43)$

4. Find the value(s) of $x$ that satisfies each of these equations.

   $2^x = 4096$

   $3^x = 6000$

   $3^x2^x = 1234$

   $\mathrm{e}^x = 42$

5. If $\log_7(x) = 5,$ what must the value of each of the following expressions be? (Hint: this is an exercise of your effective use of the “rules of logarithms”.)

   $\displaystyle \log_7\biggl(\frac{x^3}{7}\biggr)$

   $\displaystyle \log_7\bigl(49x^2\bigr)$

   $\displaystyle \log_7\biggl(\frac{\sqrt{x}}{343}\biggr)$

6. If $\log_5(x) = 3$ and $\log_5(y) = 10,$ what must the value of each of the following expressions be?

   $\displaystyle \log_5\biggl(\frac{625x}{y^2}\biggr)$

   $\displaystyle \log_5\Bigl(x\sqrt{5y}\Bigr)$

   $\displaystyle \frac{\ln\bigl(x^3y\bigr)}{\ln(25)}$

7. Rewrite each of the following expressions as a single logarithm. I.e. “simplify” these into a form that looks like $\log_b(\text{stuff})$ for some $\text{stuff}.$

   $\displaystyle \log_7(x) + 4\log_7(y)$

   $\displaystyle \ln(5x) - 3\ln(z)$

   $\displaystyle \frac{1}{2}\log_5(4x) + 2\log_5(x+1)$

8. Rewrite each of the following expressions as a single logarithm.

   $\displaystyle -3\log_2(3x) + \sqrt{5}\log_2(y)$

   $\displaystyle \frac{3\ln(w) - \frac{1}{3}\ln(y)}{\ln(3)}$

   $\displaystyle \frac{2}{3}\log_9(x) - 2\log_9(y-1) + 2\log_9(9)$

9. In May 2008 an earthquake of magnitude 6.8 struck Honshu, Japan, causing some injuries and building damage. In March 2011 a magnitude 9.0 earthquake stuck Honshu, killing thousands of people. How much more powerful was this second earthquake?