Answer:

II.

Represent each of the following situation on logarithmic function with an

equation

1.An earthquake monitoring station measured the amplitude of the waves during a

recent tremor It measured the waves as being 100,000 times as large as As the

smallest detectable wave Express in a logarithmic function that shows how high did

this earthquake measure on the Richter scale

2. One hot water pump has a noise rating of 50 decibels. One dishwasher, however, has

a noise rating of 62 decibels. Express in a logarithmic function that shows the

dishwasher noise is how many times more intense than the hot water pump noise?

Answer:

Much of the power of logarithms is their usefulness in solving exponential equations. Some examples of this include sound (decibel measures), earthquakes (Richter scale), the brightness of stars, and chemistry (pH balance, a measure of acidity and alkalinity).

Let’s look at the Richter scale, a logarithmic function that is used to measure the magnitude of earthquakes. The magnitude of an earthquake is related to how much energy is released by the quake. Instruments called seismographs detect movement in the earth; the smallest movement that can be detected shows on a seismograph as a wave with amplitude A0.

A – the measure of the amplitude of the earthquake wave

A0 – the amplitude of the smallest detectable wave (or standard wave)

From this you can find R, the Richter scale measure of the magnitude of the earthquake using the formula:

The intensity of an earthquake will typically measure between 2 and 10 on the Richter scale. Any earthquakes registering below a 5 are fairly minor; they may shake the ground a bit, but are seldom strong enough to cause much damage. Earthquakes with a Richter rating of between 5 and 7.9 are much more severe, and any quake above an 8 is likely to cause massive damage. (The highest rating ever recorded for an earthquake is 9.5 during the 1960 Valdivia earthquake in Chile.)

Example

Problem

An earthquake is measured with a wave amplitude 392 times as great as A0. What is the magnitude of this earthquake using the Richter scale, to the nearest tenth?

Use the Richter scale equation.

Since A is 392 times as large as A0, A = 392A0. Substitute this expression in for A.

R = log 392

R = 2.5932…

R 2.6

Simplify the expression

.

Use a calculator to evaluate the logarithm.

Answer

The magnitude of this earthquake is 2.6 on the Richter scale.

A difference of 1 point on the Richter scale equates to a 10-fold difference in the amplitude of the earthquake (which is related to the wave strength). This means that an earthquake that measures 3.6 on the Richter scale has 10 times the amplitude of one that measures 2.6.

Let’s look back at the example just shown. In that example, the wave amplitude of the earthquake was 392 times normal. What if it were 10 times that, or 3,920 times normal? To find the measurement of that size earthquake on the Richter scale, you find log 3920. A calculator gives a value of 3.5932…or 3.6, when rounded to the nearest tenth. One extra point on the Richter scale can mean a lot more shaking!

Sound is measured in a logarithmic scale using a unit called a decibel. The formula looks similar to the Richter scale:

where P is the power or intensity of the sound and P0 is the weakest sound that the human ear can hear.

Example

Problem

One hot water pump has a noise rating of 50 decibels. One dishwasher, however, has a noise rating of 62 decibels. The dishwasher noise is how many times more intense than the hot water pump noise?

You can’t easily compare the two noises using the formula, but you can compare them to P0. Start by finding the intensity of noise for the hot water pump. Use h for the intensity of the hot water pump’s noise.

Divide the equations by 10 to get the log by itself.

Rewrite the equation as an exponential equation.

h = 105P0

Multiply by P0 to get h by itself.

Repeat the same process to find the intensity of the noise for the dishwasher.

To compare d to h, you can divide. (Think: if the dishwasher’s noise is twice as intense as the pump’s, then d should be 2h—that is, should be 2.)

Use the laws of exponents to simplify the quotient.

Answer

The dishwasher’s noise is 101.2 (or about 15.85) times as intense as the hot water pump.

With decibels, every increase of 10 means the sound is 10 times more intense. An increase of 20 would be 10 times more intense for the first 10, and another 10 times more intense for the second 10—so a sound that is 75 decibels is 100 times more intense than a sound that is 55 decibels!

Here’s one more example of logarithms used in scientific contexts. The measure of acidity of a liquid is called the pH of the liquid. This is based on the amount of hydrogen ions (H+) in the liquid. The formula for pH is:

pH = −log[H+]

where [H+] is the concentration of hydrogen ions, given in a unit called mol/L (“moles per liter”; one mole is 6.022 x 1023 molecules or atoms).

Liquids with a low pH (down to 0) are more acidic than those with a high pH. Water, which is neutral (neither acidic nor alkaline, the opposite of acidic) has a pH of 7.0.

Example

Problem

If lime juice has a pH of 1.7, what is the concentration of hydrogen ions (in mol/L) in lime juice, to the nearest hundredth?

pH = −log[H+]

Use the formula for pH.

1.7 = −log x

Substitute the known pH into the formula, and represent H+ with the variable x.

−1.7 = log x

If 1.7 = −log x, then log x = −1.7.

x = 10-1.7

x = 0.02.

Solve for x.

Answer

The concentration of hydrogen ions in lime juice is 0.02.