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study help
mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Solve each of the given systems of equations algebraically.y = 4x2y = 8x
Solve the given systems of equations graphically by using a calculator. Find all values to at least the nearest 0.1.x² − 4y² = 16x² + y² = 1
Solve the given systems of equation algebraically.xy = 412x − 9y = 6
Solve the given equation algebraically.8n1/2 − 20n1/4 = 12
Solve the given equation.5√s − 6 = s
Solve the given systems of equations by use of a calculator.y = 2 ln xy =sin x
Solve the given systems of equations graphically by using a calculator. Find all values to at least the nearest 0.1.y = x³2x² + 4y² = 32
Solve the given systems of equations algebraically.xy = 3y − x = 2
Solve the given equation algebraically.y3 + 2y3/2 = 80
Solve the given equation.e2x = 5
If f(x) = logb x, expressas a single logarithm. f(x +h)-f(x) h
Many exponential and logarithmic equations cannot be solved algebraically as we did in this section. However, they can be solved graphically as was shown in the check of Example 8. In Exercises, solve the given equations graphically by use of a calculator.4x + x2 = 25Data from Example 8Solve the
Use a calculator to display the graphs (on the same screen) of y = loge (x2 + c), with c = −4, 0, 4. Describe the results.
Many exponential and logarithmic equations cannot be solved algebraically as we did in this section. However, they can be solved graphically as was shown in the check of Example 8. In Exercises, solve the given equations graphically by use of a calculator.2x + 3x = 50Data from Example 8Solve the
Solve the given equation.log4 z + log4 6 = log412
Using a calculator, solve the equation log10 x = x − 2.
Simplify: logb (1 + b2x) − logb (1 + b−2x)
Solve the given equation.2x/31−x = 12x+1
On a calculator, display the graphs of y1 = 2 log10 x and y2 = log10 x2. Describe any similarities or differences.
Is it true that logb (ab)x = xlogb a + x?
The atmospheric pressure P (in bars) at an altitude h (in km) above Earth’s surface can be estimated using the equation P = e−h/7. Solve for h as a function of P. At what elevation is the pressure 0.35 bars?
Solve the given equation.3x+2 = 5x
On a calculator, display the graphs of: (a) y1 = log10 x,(b) y2 = log10 (x + 2) (c) y3 = log10 (x − 2)
In an electric circuit containing a resistor and a capacitor with an initial charge q0, the charge q on the capacitor at any time t after closing the switch can be found by solving the equationHere, R is the resistance, and C is the capacitance. Solve for q as a function of t. Ing = t RC + Ingo.
If logb x = 2 and logb y = 3, find log √x2 y4 .
An Earth satellite loses 0.1% of its remaining power each week. An equation relating the power P, the initial power P0, and the time t (in weeks) is ln P = t ln 0.999 + ln P0. Solve for P as a function of t.
Solve the given equation.2(5x) = 15
Find b such that the point (8, −3) is on the graph of y = logb x.
If x = logb 2 and y = logb 3, express logb 12 in terms of x and y.
Find b such that the point (2, 2) is on the graph of y = logb x.
Display the graphs of y = loge (e2x) and y = 2 + loge x on a calculator and explain why they are the same.
Studies have shown that the concentration c (in mg/cm3 of blood) of aspirin in a typical person is related to the time t (in h) after the aspirin reaches maximum concentration by the equation ln c = ln15 − 0.20t. Solve for c as a function of t.
Use the change-of-base formula to find each logarithm.log5.02 (852.1)
If f(x) = logb x and f(3) = 2, find f(9).
Referring to Example 5, show that if the difference in loudness of two sounds is d decibels, the louder sound is 10d /10 more intense than the quieter sound.Data from Example 5The human ear responds to sound on a scale that is approximately proportional to the logarithm of the intensity of the
Express as the logarithm of a single quantity: 2 log2 (2x) − log2 x2. For what values of x is the value of this expression valid? Explain.
How many more times intense was the 1906 San Francisco earthquake (R = 8.3) than the 1935 Timiskaming earthquake (felt in Ontario, Quebec, and northeast United States) (R = 6.1)?
Use the change-of-base formula to find each logarithm.log4.25 (0.0067)
Perform the indicated operations.If f(x) = log5 x, find: (a) f(√5) (b) f(0)
Explain why log10 (x + 3) is not equal to log10 x + log10 3.
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. How many times I0 was the 1906 San Francisco earthquake whose magnitude was 8.3 on the Richter scale?
Use the change-of-base formula to find each logarithm.log7 (0.019)
Evaluate: (a) logb b (b) logb 1
Solve for y in terms of x.log8 x = 2 log8 y + 4
Use the change-of-base formula to find each logarithm.log3 (594)
Evaluate log4 x for (a) x = 1/64 (b) x = −1/2.
Solve for y in terms of x.2 log5 x − log5 y = 2
In chemistry, the pH value of a solution is a measure of its acidity. The pH value is defined by pH = −log(H+), where H+ is the hydrogen-ion concentration. If the pH of a sample of rainwater is 4.764, find the hydrogen-ion concentration. (If pH < 7, the solution is acid. If pH > 7, the
The electric current i (in A) in a circuit that has a 1-H inductor, a 10-Ω resistor, and a 6-V battery, and the time t (in s) are related by the equation 10t = −ln(1 − i/0.6). Solve for i.
The heat loss rate Q (in W/m) through a certain cylindrical pipe insulation is given bywhere T1 is the temperature inside the insulation, T2 is the temperature outside the insulation, and r1 and r2 are the inside and outside radii of the insulation, respectively. Find the heat loss rate if T1
Use a calculator to find each logarithm.ln(0.0052)
Display the graphs of the given functions on a graphing calculator.y = −loge (−2x)
Solve for y in terms of x.πlog4 x + log4 y = 1
When a camera flash goes off, the batteries recharge the flash’s capacitor to a charge Q according to Q = Q0 (1 − e−kt), where Q0 is the maximum charge. How long does it take to recharge the capacitor to 90% of capacity if k = 0.5?
For the electric circuit of Example 6, find how long it takes the current to reach 0.1 of the initial value of 0.750 A.Data from Example 6The electric current i in a circuit containing a resistance and an inductance (see Fig. 13.18) is given by ln(i/I ) = −Rt/L, where I is the current at t = 0, R
The distance x traveled by a motorboat in t seconds after the engine is cut off is given by x = k−1 ln(kv0t + 1), where v0 is the velocity of the boat at the time the engine is cut and k is a constant. Find how long it takes a boat to go 150 m if v0 = 12.0 m/s and k = 6.80 ×10−3 /m.
Use a calculator to find each logarithm.ln(87.9)
Display the graphs of the given functions on a graphing calculator.y = −log10 (−x)
Solve for y in terms of x.log2 x + log2 y = 1
Forensic scientists determine the temperature T (in °C) of a body t hours after death from the equation T = T0 + (37 − T0)0.97t , where T0 is the air temperature. If a body is discovered at midnight with a body temperature of 27°C in a room at 22°C, at what time did death occur?
Use a calculator to find each logarithm.log(3.16 ×105)
Display the graphs of the given functions on a graphing calculator.y = 5 log10 |x|
Solve for y in terms of x.4 log2 x − 3 log2 y = log2 27
Referring to Exercise 66 in Section 13.3, in what year will the amount of DDT be 25% of the original amount?Data from Exercises 66 in Section 13.3:The use of the insecticide DDT was banned in the United States in 1972. A computer analysis shows that an expression relating the amount A still present
World population is currently growing by 1.1% annually. If it continues at this rate, the time (in years) for the population to double in size is given byHow many years is this (to the nearest year)? t = In 2 In 1.011
Use a calculator to find each logarithm.log(0.0824)
Display the graphs of the given functions on a graphing calculator.y = 3 loge x
Solve for y in terms of x.5 log2 y − log2 x = 3 log2 4 + log2 a
According to one model, the number N of Americans (in millions) age 65 and older that will have Alzheimer’s disease t years after January 1, 2020 is given by N = 5.8(1.03)t . In what year will this number reach 8.0 million?
Display the graphs of the given functions on a calculator.s = 2(1 − e−0.2t)
Plot the graphs of the given functions.A = 2.4 log10 (2r)
Solve for y in terms of x.logb y = 3 logb √x + 2 logb 10
Use a calculator to find the point of intersection of the curves of 3x + 5y = 6 and y = 1.5 ln(x + 3).
If interest is compounded continuously with an interest rate i, a bank account will double in t years according to i = (ln 2)/t. Find i if the account is to double in 25 years.
Display the graphs of the given functions on a calculator.y = 1 − e−|x|
Plot the graphs of the given functions.N = 0.2 log4 v
Use logarithms to find the x-intercept of the graph of y = 3 − 4x+2.
Solve for y in terms of x.log10 y = 2 log10 7 − 3 log10 x
Find k(in 1/Pa) if ln k = −21.504, where k is the compressibility of water.
Display the graphs of the given functions on a calculator.s = 0.1 log4.05 t
Plot the graphs of the given functions.y = 3 log2 x
What values of x cannot be solutions of the equation y = log(2x − 5) + log(x2 + 1)?
Find f (in Hz) if ln f = 21.619, where f is the frequency of the microwaves in a microwave oven.
Display the graphs of the given functions on a calculator.y = log3.15 x
Plot the graphs of the given functions.y = log0.5 x
If y = 1.5e−0.90x, find y when x = 7.1.
Solve for y in terms of x.log4 y = log4 x − log4 10 + log4 6
Expressas the sum or difference of logarithms, evaluating where possible. In(e²√1-x)
Simplify: 2 ln(e3x+1) − eln(2x−3) .
Display the graphs of the given functions on a calculator.y = 10 log16 x
The height y (in m) of the Gateway Arch in St. Louis (see Fig. 13.5) is given by y = 230.9 − 19.5(e x/38.9 + e−/38.9), where x is the distance (in m) from the point on the ground level directly below the top. Display the graph on a calculator.
Plot the graphs of the given functions.y = −log4(−x)
Solve for x: 3x + 3−x = 4. See Exercise 45.Data from Exercises 45Solve for x: ex + e−x = 3.
Solve for y in terms of x.logb y = logb 6 − logb x
Display the graphs of the given functions on a calculator.R = 0.2 log4 r
The flash unit on a camera operates by releasing the stored charge on a capacitor. For a particular unit, the charge q (in μC) as a function of the time t (in s) is q = 100e−10t. Display the graph on a calculator.
Plot the graphs of the given functions.y = log3x
Solve for x: ex + e−x = 3.
Solve for y in terms of x.logb y = logb 2 + logb x
The electric current i (in mA) in the circuit shown in Fig. 13.4 is i = 2.5(1 − e−0.10t), where t is the time (in s). Evaluate i for t = 5.0ms (1ms = 10−3 s). 10 Ω www 50 V L 10 Ω Fig. 13.4 2 μF
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