A function f(x)=ax, with a>0 and a≠1, is called an exponential function with base a and exponent x.
Rules of exponents: axay=ax+y,axay=ax−y,
(ab)x=axbx,(ab)x=axbx(ax)y=axy,a0=1,a−x=1ax
Exponential functions are one-to-one: If au=av, then u=v.
If a>1, then f(x)=ax is an increasing function; f(x)→∞ as x→∞ and f(x)→0 as x→−∞.
If 0<a<1, then f(x)=ax is a decreasing function; f(x)→0 as x→∞ and f(x)→∞ as x→−∞.
The graph of f(x)=ax has y-intercept 1, and the x-axis is a horizontal asymptote.
Simple interest formula. If P dollars is invested at an interest rate r per year for t years, then the simple interest is given by the formula I=Prt. The future value A(t)=P+Prt.
Compound interest.P dollars invested at an annual rate r compounded n times per year for t years amounts to
A(t)=P(1+rn)nt.
The Euler constant e=limh→∞(1+1h)h≈2.718.
Continuous compounding.P dollars invested at an annual rate r compounded continuously for t years amounts to A=Pert.
The function f(x)=ex is the natural exponential function.
f(x)=3x is an exponential function with base 3 and exponent x: f(2)=32=9.
If 3x=34, then x=4.
$500 is invested at 3% for 6 years. Find the amount A if the interest is compounded continuously.
Solution
We use A=Pert.
Here P=$500,r=3%=0.03, and t=5.
So,
A==500e(0.03)(5)500e0.15≈$580.92
4.2 Logarithmic Functions
For x>0,a>0, and a≠1,y=logax if and only if x=ay.
Basic properties: logaa=1,loga1=0,
logaax=x,alogax=xInverse properties
The domain of logax is (0,∞), the range is (−∞,∞), and the y-axis is a vertical asymptote. The x-intercept is 1.
Logarithmic functions are one-to-one: If logax=logay, then x=y.
If a>1, then f(x)=logax is an increasing function; f(x)→∞ as x→∞ and f(x)→−∞ as x→0+.
If 0<a<1, then f(x)=logax is a decreasing function; f(x)→−∞ as x→∞ and f(x)→∞ as x→0+.
The common logarithmic function is y=logx (base 10); the natural logarithmic function is y=lnx (base e).
log57=log37log35=≈log7log5=ln7ln51.2091Change of baseUse a calculator.
4.4 Exponential and Logarithmic Equations and Inequalities
An exponential equation is an equation in which a variable occurs in one or more exponents.
A logarithmic equation is an equation that involves the logarithm of a function of the variable.
Exponential and logarithmic equations are solved by using some or all of the following techniques:
Using the one-to-one property of exponential and logarithmic functions.
Converting from exponential to logarithmic form or vice versa
Using the product, quotient, and power rules for exponents and logarithms
We use similar techniques to solve logarithmic and exponential inequalities (see page 483).
Solve: 23x−1=32.
Solution
23x−13x−13xx====25562.32=251−1 propertyAdd 1 to both sides.Divide both sides by 3.
Solve:log(x2−2x)=log8.
Solution
x2−2xx2−2x−8(x+2)(x−4)===8001−1 property of logarithmSubtract 8 from both sides.Factor.x+2x==0−2ororx−4x==04Zero-product property.Solve for x.
Both values check. Solution set is {−2,4}.
Solve:3x=13.
Solution
log3xxlog3x===≈log13log13log13log32.3347Take log of both sides.Power ruleDivide by log3.Exact Solution.Use a calculator.
4.5 Logarithmic Scales
A logarithmic scale is a scale in which logarithms are used in the measurement of quantities.
pH=−log[H+], where [H+] is the concentration of H+ ions in moles per liter.
The Richter scale is used to measure the magnitudeM of an earthquake.
M=log(II0), where I is the intensity of the earthquake and I0 is the zero-level earthquake.
The energyE released by an earthquake of magnitude M is given by logE=4.4+1.5M.
The intensity of a sound wave is defined as the amount of power the wave transmits through a given area. The loudnessL of a sound of intensity I is given by L=10log(II0), where I0=10−12W/m2 is the intensity of the threshold of hearing. Equivalently, I=10L/10I0.