Calculus I - Logarithm Functions (Practice Problems)

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Section 1.8 : Logarithm Functions

Without using a calculator determine the exact value of each of the following.

${\log _3}81$ Solution
${\log _5}125$ Solution
$\displaystyle {\log _2}\frac{1}{8}$ Solution
$\displaystyle {\log _{\frac{1}{4}}}16$ Solution
$\ln {{\bf{e}}^4}$ Solution
$\displaystyle \log \frac{1}{{100}}$ Solution

Write each of the following in terms of simpler logarithms.

$\log \left( {3{x^4}{y^{ - 7}}} \right)$ Solution
$\ln \left( {x\sqrt {{y^2} + {z^2}} } \right)$ Solution
$\displaystyle {\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right)$ Solution

Combine each of the following into a single logarithm with a coefficient of one.

$\displaystyle 2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z$ Solution
$3\ln \left( {t + 5} \right) - 4\ln t - 2\ln \left( {s - 1} \right)$ Solution
$\displaystyle \frac{1}{3}\log a - 6\log b + 2$ Solution

Use the change of base formula and a calculator to find the value of each of the following.

${\log _{12}}35$ Solution
${\log _{\frac{2}{3}}}53$ Solution