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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Express ln(xe−x) as the sum or difference of logarithms, evaluating where possible.
Display the graphs of the given functions on a calculator.y = 3(2−x)
The strength I of a certain cable signal is given by I = I0e−0.0015x , where I0 is the signal strength at the source and x is the distance (in km) from the source. What percent of the signal strength is lost 15 km from the source?
Determine the value of the unknown.logb (1/4) = −0.5
If 2−x = 7, find y if y = 42x.
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.3 log10 (402)
A certain type of optical switch in a fiber-optic system allows a light signal to continue in either of two fibers. How many possible paths could a light signal follow if it passes through 400 such switches?
If x = ln 4 and y = ln 5, express ln 80 in terms of x and y.
Display the graphs of the given functions on a calculator.y = 0.5(5x)
Determine the value of the unknown.log3 27−1 = x + 1
If 4x = 5, find y if y = 4−3x.
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log10 3000
How many more times was the intensity of the 2011 earthquake in Japan (R = 9.0) than the 2015 earthquake in Nepal (R = 7.8)? (See Exercise 41.)Data from Exercises 41Measured on the Richter scale, the magnitude of an earthquake of intensity I is defined as R = log(I/I0), where I0 is a minimum level
If ln x = 3 and ln y = 4, find √x2y.
Solve for y in terms of x.10y = 3x+1
The intensity I of an earthquake is given by I = I0(10) R, where I0 is a minimum intensity for comparison and R is the Richter scale magnitude of the earthquake. Evaluate I in terms of I0 if R = 5.5.
Determine the value of the unknown.log5 52.3 = R + 1
Use a calculator to solve the given equation.9x − 3x − 12 = 0
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log2 3√24
Measured on the Richter scale, the magnitude of an earthquake of intensity I is defined as R = log(I/I0), where I0 is a minimum level for comparison. What is the Richter scale reading for the 1964 Alaska earthquake for which I = 1,600,000,000 I0?
Solve for x: ln(log x) = 0.
Solve for y in terms of x.logx y = ln e3
The value V of a bank account in which $750 is invested at 2% interest, compounded annually, is V = 750(1.02)t , where t is the time in years. Find the value of the account after 4 years to the nearest penny.
Determine the value of the unknown.log10 100.2 = x
Use a calculator to solve the given equation.22x − 2x − 6 = 0
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log3 √6
In a particular electric circuit, called a low-pass filter, the input voltage Vi is across a resistor and a capacitor, and the output voltage V0 is across the capacitor (see Fig. 13.28). The voltage gain G (in dB) is given byHere, ∅ is the phase angle of V0/Vi. For values of ωT of 0.01, 0.1,
A stereo amplifier has an input power of 0.750 W and an output power of 25.0 W. What is the power gain? (See Example 4.)Data from Example 4The power gain G [in decibels (dB)] of an electronic device is given by G = 10 log(P0/Pi), where P0 is the output power (in W) and Pi is the input power.
Solve for y in terms of x: ln y + 2 ln x = 1 + ln 5.
A medical research lab is growing a virus for a vaccine that grows at a rate of 2.3% per hour. If there are 500.0 units of the virus originally, the amount present after t hours is given by N = 500.0(1.023)t. How many units of the virus are present after two days?
Determine the value of the unknown.3 logb 4 = 2
Use a calculator to solve the given equation.log (x − 3) + log x = log 4
In a particular electric circuit, called a low-pass filter, the input voltage Vi is across a resistor and a capacitor, and the output voltage V0 is across the capacitor (see Fig. 13.28). The voltage gain G (in dB) is given byHere, ∅ is the phase angle of V0/Vi. For values of ωT of 0.01, 0.1,
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log10 (0.05)
Solve the given problems by evaluating the appropriate logarithms.Evaluate: 2(10log 0.1) + 3(10log 0.01).
Use a calculator to solve the given equation.3 ln 2x = 2
Evaluate: √ln e9.
For f (x) = Aekx, find the expression for f(x + 1) − f(x) f(x)
The magnetic intensity H (in A/m) and flux density B (in teslas) of annealed iron are given in the following table. Plot H as a function of B on log-log paper. B (T) H (A/m) B (T) H (A/m) 0.0042 0.043 10 50 0.67 0.67 1.01 100 150 1.18 1.44 1.58 1.72 200 500 1000 10,000
Determine the value of the unknown.logb 4 = −1/3
Use a calculator to solve the given equation.2 ln 2 − ln x = −1
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log2 (4/7)
Solve the given problems by evaluating the appropriate logarithms.Evaluate: log(log 10100) (10100 is called a googol.)
Solve the given problems by evaluating the appropriate logarithms.Simplify: 2 log, x² log 100
Express as the logarithm of a single quantity.3 ln x − 2 ln z
One end of a very hot steel bar is sprayed with a stream of cool water. The rate of cooling R (in °F/s) as a function of the distance d (in in.) from one end of the bar is then measured, with the results shown in the following table. On log-log paper, plot R as a function of d. Such experiments
Solve for y in terms of x.log7 y = 2 log7 5 + log7 x + 3
Use a calculator to find the value(s) for which x3 = 3x.
Determine the value of the unknown.logb 625 = 4
Use a calculator to solve the given equation.log 4x + log x = 2
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log5 375
Express as the logarithm of a single quantity.4 ln 3 + ln y
Solve for y in terms of x.log5 x + log5 y = log5 3 + 1
Use a calculator to find the value(s) for which x2 = 2x.
Determine the value of the unknown.logb 3 = 2
Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly.log3 18
Determine the exact value of each of the given expressions.102 log103
Solve for y in terms of x.2(log4 y − 3 log4 x) = 3
Display the graphs of the given functions on a calculator.(a) y = 10x (b) y = 2x (c) y = 1.1x
The intensity level B (in dB) and the frequency f (in Hz) for a sound of constant loudness were measured as shown in the table that follows. Plot the data for B as a function of f on semilog paper, using the log scale for f. f(Hz) 100 200 200 B (dB) 40 30 500 500 1000 2000 5000 22 20 18 24 10,000 30
Find the indicated values.Find e if log e = −0.35, where e is the efficiency of a certain gasoline engine.
The atmospheric pressure p (in kPa) at a given altitude h (in km) is given in the following table. On semilog paper, plot p as a function of h. h (km) P (Pa) 0 10 20 30 40 101 25 6.3 2.0 0.53
Find E (in J) if log E = −18.49, where E is the energy of a photon of visible light.
Express as the logarithm of a single quantity.2 ln x − ln 3
Solve for y in terms of x.(log2 3)(log2 y) − log2 x = 3
To show the damping effect of an exponential function, use a calculator to display the graph of y = (x3)(2−x). Be sure to use appropriate window settings.
Determine the value of the unknown.log7 y = −2
Use a calculator to solve the given equation.5x+2 = 3e2x
The period T (in years) and mean distance d (given as a ratio of that of Earth) from the sun to the planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto) are given below. Plot T as a function of d on log-log paper. MVE M J S UN P 19.2 30.1 39.5 0.39 0.72 1.00 1.52 5.20
Determine the exact value of each of the given expressions.4log48
Express as the logarithm of a single quantity.ln 5 + ln y + ln t
Solve for y in terms of x.log3 y = 1/2 log3 7 + 1/2 log3 x
Use a calculator to graph the function y = 2|x|.
Determine the value of the unknown.3 log8 (A − 2) = −2
Use a calculator to solve the given equation.4(3x) = 5
Determine the exact value of each of the given expressions.πlog6 3√6
The number N of daily new cases of COVID-19 that were reported in the United States t weeks after March 1, 2019 is shown in the following table. Plot N as a function of t on semilog paper. t 0 N 16 1 198 2 1261 3 9109 4 19,303
Find v (in m/s) if log v = 8.4768, where v is the speed of light.
Explain what is meant by the expression ln ln x. Display the graph of y = ln ln x on a calculator.
Solve for y in terms of x.2(log9 y + 2 log9 x) = 1
For the function f(x) = bx , show that f(c − d) = f(c)/f(d).
Graphically show that y = 2x and y = log2 x are inverse functions. (See Example 9 of Section 13.2.)Data from Example 9 of Section 13.2.The functions y = 2x and y = log2 x are inverse functions. We show this by solving y = 2x for x and then interchanging x and y. Writing y = 2x in logarithmic form
Determine the value of the unknown.log8(N + 1) = 3
Use a calculator to solve the given equation.e2x = 3.625
Determine the exact value of each of the given expressions.6 log7 √7
Find T (in K) if log T = 8, where T is the temperature sufficient for nuclear fission.
Solve for y in terms of x.3 ln y = 2 + 3 ln x
For the function f(x) = bx , show that f (c + d) = f(c) · f(d).
Use a calculator to solve the given equation.15−x = 1.326
Determine the exact value of each of the given expressions.log5 (50.1)
Determine the value of the unknown.log7 y = 3
In undergoing an adiabatic (no heat gained or lost) expansion of a gas, the relation between the pressure p (in kPa) and the volume v (in m3 ) is p2v3 = 850. On log-log paper, graph p as a function of v from v = 0.10 m3 to v = 10 m3.
Find the logarithms of the given numbers. In an air sample taken in an urban area, 0.000005 of the air was carbon dioxide.
Use a calculator to display the indicated graphs.The graph of y = 2 log8 x
Solve for y in terms of x.6 ln y = 3ln e2 − 9 ln x
Find the base b of the function y = b x if its graph passes through the point (−2, 64).
Determine the value of the unknown.log16(1/4) = x
Solve the given equation.ln(2x − 1) − 2 ln 4 = 3ln 2
The electric power P (in W) in a certain battery as a function of the resistance R (inΩ) in the circuit is given byPlot P as a function of R on semilog paper, using the logarithmic scale for R and values of R from 0.01Ω to 10Ω. P = 100R (0.50 + R)2
Determine the exact value of each of the given expressions.log2 (22.5)
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