The pH of a solution is modeled by \(\text{pH}=-\log_{10}([H^+])\), where \([H^+]\) means the numerical hydrogen-ion concentration in \(\text{mol}/\text{L}\). Find the pH when \([H^+]=10^{-4}\).

Sound level is modeled by \(L=10\log_{10}\left(\frac{I}{I_0}\right)\), where \(I_0\) is a reference intensity. If \(L=20\,\text{dB}\), what is the ratio \(\frac{I}{I_0}\)?

Is this measurement axis linear or logarithmic? If it is logarithmic, what factor does one equal interval represent? Explain.

The vertical axis uses a logarithmic scale. Find the value at \(P\), and tell how much the value is multiplied by when you move up one tick.

Consider positive integers written in base \(10\). a) What do \(\log(n)\) and \(\log(m)\) have in common when \(n\) and \(m\) have the same number of digits? Here \(\log\) denotes the common logarithm. b) In what interval does \(\log(x)\) lie when \(x\) is any six-digit positive integer? Write the interval in the form \([a, b)\).

A logarithmic index is defined by \(S(r)=10\log_{10}(r)\) for a positive ratio \(r\). a) How much does \(S\) increase when \(r\) changes from \(1\) to \(10\)? How much does it increase from \(10\) to \(100\)? b) Suppose \(r\) is plotted on an ordinary linear horizontal axis. Compare the input changes \(1\to10\) and \(10\to100\). Explain how the logarithmic index compresses multiplicative changes. c) Predict \(S(1000)\) without evaluating a logarithm on a calculator.

On this base-10 logarithmic axis: a) What value does \(p\) represent? Give an exact expression and a decimal approximation to the nearest tenth. b) Explain why averaging the two neighboring labeled values does not give \(p\).

An earthquake-amplitude scale uses \(M=\log_{10}\!\left(\frac{A}{A_0}\right)\), where \(A\) is the measured amplitude and \(A_0\) is a reference amplitude. One quake has \(M=2\) and another has \(M=4\). Find each amplitude as a multiple of \(A_0\), and find the ratio of their amplitudes.

The pH of a liquid is given by \(\text{pH}=-\log_{10}(c)\), where \(c\) is the numerical hydrogen-ion concentration in moles per liter. Find \(c\) for liquids with pH \(5\) and pH \(6\). How many times as concentrated is the first liquid?

A simplified brightness scale for stars is \(m=-2.5\log_{10}\left(\frac{F}{F_0}\right)\), where \(F\) is a star's measured brightness and \(F_0\) is a fixed reference. Star A has \(F=100F_0\), and star B has \(F=F_0\). Find the magnitude of each star. Which appears brighter on this scale?

In a simplified earthquake-energy model, \(\log_{10}\left(\frac{E}{E_0}\right)=1.5M\), where \(E\) is released energy, \(E_0\) is a fixed reference, and \(M\) is magnitude. How many times as much energy does a magnitude-\(5\) earthquake release as a magnitude-\(3\) earthquake according to this model?

The vertical logarithmic scale has the same multiplication factor between each pair of neighboring ticks. Find the value at \(A\), then find the ratio of the values at \(B\) and \(A\).

Determine the number of digits in the decimal representation of \(6^{60}\). Use \(\log(2)\approx0.3010\) and \(\log(3)\approx0.4771\).

A computer program can store whole numbers with at most \(100\) decimal digits. Find the greatest natural number \(n\) for which \(2^n\) can still be stored. Use \(\log(2)\approx0.30103\).

The magnitude \(M\) of an earthquake can be modeled from its released seismic energy \(E\), in joules, by \(M=\frac{2}{3}\log_{10}\!\left(\frac{E}{E_0}\right)\), where \(E_0=10^{4.8}\,\text{J}\). How many times as much energy is released by an earthquake of magnitude \(7.5\) as by an earthquake of magnitude \(5.5\)?

The Weber-Fechner model for perceived stimulus strength is \(S(I)=k\ln\!\left(\frac{I}{I_0}\right)\), where \(I\) is physical intensity, \(I_0\) is the threshold intensity, and \(k\) is a constant. 1. For sound perception, suppose \(I_0=10^{-12}\,\text{W/m}^2\). At \(I_1=10^{-7}\,\text{W/m}^2\), the measured perception strength is \(S_1=50\). Find \(k\). 2. Show algebraically that doubling the intensity always produces the same absolute change in perceived strength, regardless of the starting intensity. 3. Using the value of \(k\) from part 1, find the absolute change in perceived strength when intensity is multiplied by \(10\).

In acoustics, sound intensity and sound level are related logarithmically. The table shows several values. <table> <tr> <td>Sound intensity \(I\) (in \(\text{W}/\text{m}^2\))</td> <td>\(10^{-12}\)</td> <td>\(10^{-10}\)</td> <td>\(10^{-8}\)</td> <td>\(10^{-6}\)</td> </tr> <tr> <td>Sound level \(L\) (in \(\text{dB}\))</td> <td>\(0\)</td> <td>\(20\)</td> <td>\(40\)</td> <td>\(60\)</td> </tr> </table> In the formula below, \(I\) means the numerical intensity reading in \(\text{W}/\text{m}^2\), and \(L\) is the numerical sound-level reading in decibels. a) By how many decibels does \(L\) increase when \(I\) is multiplied by \(10\)? b) The relationship can be modeled by \(L=a\log(I)+b\). Find \(a\) and \(b\), and verify the model using another pair from the table. c) Find the inverse formula for intensity in \(\text{W}/\text{m}^2\) as a function of \(L\). What intensity corresponds to \(50\,\text{dB}\)?

A digital light sensor measures illuminance in lux (\(\text{lx}\)) and produces a voltage in millivolts (\(\text{mV}\)). Let \(x\) be the numerical illuminance reading in lux and \(U\) the numerical voltage reading in millivolts. The sensor has a logarithmic response. <table> <tr> <td>Illuminance \(x\) (in \(\text{lx}\))</td> <td>\(1\)</td> <td>\(10\)</td> <td>\(100\)</td> <td>\(1000\)</td> </tr> <tr> <td>Voltage \(U\) (in \(\text{mV}\))</td> <td>\(5\)</td> <td>\(25\)</td> <td>\(45\)</td> <td>\(65\)</td> </tr> </table> a) How does \(U\) change when \(x\) is multiplied by \(10\)? b) Find a model of the form \(U=a\log(x)+b\). c) The sensor produces \(55\,\text{mV}\). Find the corresponding illuminance, rounded to the nearest hundredth of a lux. Assume the model applies between the table values.

In acoustics, the sound level \(L\), in decibels (\(\text{dB}\)), of a sound with intensity \(I\) is modeled by \(L=10\log\left(\frac{I}{I_0}\right)\), where \(I_0\) is the threshold-of-hearing intensity. a) A passing car has intensity \(I=5\times10^6I_0\). Find its sound level. Round to the nearest hundredth of a decibel. b) A sound barrier reduces traffic noise from \(84\,\text{dB}\) to \(66\,\text{dB}\). By what factor is the sound intensity reduced? Round the factor to the nearest hundredth. c) Use logarithm properties to show that a decrease of exactly \(10\,\text{dB}\) always reduces the intensity to one tenth of its original value.

Sound level \(L\), in decibels (\(\text{dB}\)), is modeled by \(L=10\log\left(\frac{I}{I_0}\right)\). a) A speaker produces a sound level of \(75\,\text{dB}\). What multiple of the threshold intensity \(I_0\) is this? b) When two identical speakers operate together, the sound intensity doubles. Find the new sound level if one speaker produces \(75\,\text{dB}\). Round to the nearest hundredth of a decibel. c) How many identical speakers must operate together to produce a total sound level of \(85\,\text{dB}\)?

In chemistry, \(\text{pH}=-\log(c)\), where \(c\) is the hydronium-ion concentration expressed as a number in \(\text{mol}/\text{L}\). Use the base-\(10\) logarithm. a) Find the pH of each liquid. (1) Stomach acid with a hydronium-ion concentration of \(0.032\,\text{mol}/\text{L}\). Round to the nearest hundredth. (2) A soap solution with a concentration of \(10^{-10}\,\text{mol}/\text{L}\). b) Soil for rhododendrons should have a pH of about \(5.5\). Find the corresponding hydronium-ion concentration in \(\text{mol}/\text{L}\). Express it in scientific notation with its coefficient rounded to the nearest hundredth. c) Lemon juice has pH \(2.4\), while vinegar has pH \(2.9\). By what factor do their hydronium-ion concentrations differ? Round to the nearest hundredth.

Sound intensity level \(L\), in decibels (\(\text{dB}\)), is calculated by \(L=10\log\left(\frac{I}{I_0}\right)\), where \(I\) is the sound intensity and \(I_0=10^{-12}\,\text{W}/\text{m}^2\) is the reference intensity at the threshold of hearing. a) A conversation-level sound has intensity about \(10^{-6}\,\text{W}/\text{m}^2\). Find its sound intensity level. b) A sound level of \(120\,\text{dB}\) is commonly associated with the threshold of pain. Find the corresponding intensity \(I\). c) If the intensity doubles, by how many decibels does the sound intensity level increase? Round to the nearest hundredth of a decibel.

The two panels show the same four marked measurements \(p,q,r,s\). One panel uses a linear horizontal scale and the other uses a base-10 logarithmic horizontal scale. Do not assume the panel order. a) Identify the logarithmic panel and justify your choice using the displayed numerical scale and marker spacing. b) Describe what happens to the spacing of \(p,q,r,s\) on the linear panel compared with the logarithmic panel. c) Which panel is more informative when the purpose is to compare repeated multiplicative changes across several orders of magnitude? Explain.

Sound level \(L\), in decibels, can be written using the natural logarithm as \(L=\frac{10}{\ln(10)}\ln\!\left(\frac{I}{I_0}\right)\), where \(I\) is sound intensity and \(I_0\) is a reference intensity. One jackhammer produces a sound level of \(85\,\text{dB}\). a) Find the total sound level when a second identical jackhammer operates beside the first. Assume the intensities add. Round to the nearest hundredth of a decibel. b) Find the minimum number of identical jackhammers that must operate simultaneously to produce a sound level of at least \(100\,\text{dB}\).