- Write down an explanation of how you can estimate the value of \(\log_{10}(31415926)\) without appealing to technology.
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Without appealing to technology,
figure out the exact value
of the following logarithms.
(Hint: they’re all whole numbers.)
\(\log_5(625)\)\(\log_2(1024)\)\(\log_{11}(14641)\)\(\ln\bigl(\mathrm{e}^7\bigr)\)
-
Using a calculator,
write down a decimal approximation of
each of the following numbers
rounded to two decimal places.
\(\log_{10}(31415926)\)\(\ln(100)\)\(\log_5(43)\)
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Find the value(s) of \(x\)
that satisfies each of these equations.
\(2^x = 4096\)\(3^x = 6000\)\(3^x2^x = 1234\)\(\mathrm{e}^x = 42\)
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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”.)
\(\displaystyle \log_7\biggl(\frac{x^3}{7}\biggr)\)\(\displaystyle \log_7\bigl(49x^2\bigr)\)\(\displaystyle \log_7\biggl(\frac{\sqrt{x}}{343}\biggr)\)
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If \(\log_5(x) = 3\) and \(\log_5(y) = 10,\)
what must the value of each of the following expressions be?
\(\displaystyle \log_5\biggl(\frac{625x}{y^2}\biggr)\)\(\displaystyle \log_5\Bigl(x\sqrt{5y}\Bigr)\)\(\displaystyle \frac{\ln\bigl(x^3y\bigr)}{\ln(25)}\)
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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 \(\text{stuff}.\)
\(\displaystyle \log_7(x) + 4\log_7(y) \)\(\displaystyle \ln(5x) - 3\ln(z) \)\(\displaystyle \frac{1}{2}\log_5(4x) + 2\log_5(x+1) \)
-
Rewrite each of the following expressions
as a single logarithm.
\(\displaystyle -3\log_2(3x) + \sqrt{5}\log_2(y) \)\(\displaystyle \frac{3\ln(w) - \frac{1}{3}\ln(y)}{\ln(3)}\)\(\displaystyle \frac{2}{3}\log_9(x) - 2\log_9(y-1) + 2\log_9(9) \)
- 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?