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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

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
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
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
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
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
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
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
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 =
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
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
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
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,
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.
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.
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
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
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
Determine the exact value of each of the given expressions.log2 (22.5)

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