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Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $b,$ and how does the notation differ?

For the following exercises, use the definition of common and natural logarithms to simplify.

$\log ({100}^{8})$

For the following exercises, use the definition of common and natural logarithms to simplify.

${10}^{\log (32)}$

For the following exercises, use the definition of common and natural logarithms to simplify.

$2\log (.0001)$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

$\log (10,000)$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

$\log (0.001)$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

$\log (1)+7$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

$2\log ({100}^{-3})$

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

$\log (0.04)$

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

$\log \left(\sqrt{2}\right)$

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula $\log \frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}$ where $M$ is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

$\log \left(\frac{1}{100}\right)=-2$

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D=10\log \left(\frac{I}{{I}_{0}}\right),$ where $I$ is the intensity of the sound in watts per square meter and ${I}_{0}={10}^{-12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of $8.3\cdot {10}^{2}$ watts per square meter?

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$1000{(1.03)}^{t}=5000$ using the common log.

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$3{(1.04)}^{3t}=8$ using the common log

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

${3}^{4x-5}=38$ using the common log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

The magnitude M of an earthquake is represented by the equation $M=\frac{2}{3}\log \left(\frac{E}{{E}_{0}}\right)$ where $E$ is the amount of energy released by the earthquake in joules and ${E}_{0}={10}^{4.4}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $1.4\cdot {10}^{13}$ joules of energy?

What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

Recall the formula for calculating the magnitude of an earthquake, $M=\frac{2}{3}\log \left(\frac{S}{{S}_{0}}\right).$ Show each step for solving this equation algebraically for the seismic moment $S.$

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

Recall the formula for calculating the magnitude of an earthquake, $M=\frac{2}{3}\log \left(\frac{S}{{S}_{0}}\right).$ One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.