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vertical asymptote : a vertical linex=ax=awhere the graph tends toward positive or negative infinity as the inputs approachaa | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
zeros : in a given function, the values ofxxat whichy=0y=0, also called roots | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
An exponential function is defined as a function with a positive constant other than11raised to a variable exponent. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
A function is evaluated by solving at a specific value. SeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
An exponential model can be found when the growth rate and initial value are known. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
An exponential model can be found when the two data points from the model are known. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
An exponential model can be found using two data points from the graph of the model. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
An exponential model can be found using two data points from the graph and a calculator. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The value of an account at any timettcan be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The numbereeis a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation iseâ2.718282.eâ2.718282. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Scientific and graphing calculators have the key[ex][ex]or[exp(x)][exp(x)]for calculating powers ofe.e.SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Continuous growth or decay models are exponential models that useeeas the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. SeeExample 11andExample 12. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The graph of the functionf(x)=bxf(x)=bxhas ay-intercept at(0,1),(0,1),domain(ââ,â),(ââ,â),range(0,â),(0,â),and horizontal asymptotey=0.y=0.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Ifb>1,b>1,the function is increasing. The left tail of the graph will approach the asymptotey=0,y=0,and the right tail will increase without bound. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
If0<b<1,0<b<1,the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptotey=0.y=0. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=bx+df(x)=bx+drepresents a vertical shift of the parent functionf(x)=bx.f(x)=bx. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=bx+cf(x)=bx+crepresents a horizontal shift of the parent functionf(x)=bx.f(x)=bx.SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Approximate solutions of the equationf(x)=bx+c+df(x)=bx+c+dcan be found using a graphing calculator. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=abx,f(x)=abx,wherea>0,a>0,represents a vertical stretch if|a|>1|a|>1or compression if0<|a|<10<|a|<1of the parent functionf(x)=bx.f(x)=bx.SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When the parent functionf(x)=bxf(x)=bxis multiplied byâ1,â1,the result,f(x)=âbx,f(x)=âbx,is a reflection about thex-axis. When the input is multiplied byâ1,â1,the result,f(x)=bâx,f(x)=bâx,is a reflection about they-axis. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
All translations of the exponential function can be summarized by the general equationf(x)=abx+c+d.f(x)=abx+c+d.SeeTable 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Using the general equationf(x)=abx+c+d,f(x)=abx+c+d,we can write the equation of a function given its description. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Logarithmic functions with basebbcan be evaluated mentally using previous knowledge of powers ofb.b.SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Common logarithms can be evaluated mentally using previous knowledge of powers of10.10.SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When common logarithms cannot be evaluated mentally, a calculator can be used. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Real-world exponential problems with base1010can be rewritten as a common logarithm and then evaluated using a calculator. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Natural logarithms can be evaluated using a calculatorExample 8. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve forx.x.SeeExample 1andExample 2 | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The graph of the parent functionf(x)=logb(x)f(x)=logb(x)has anx-intercept at(1,0),(1,0),domain(0,â),(0,â),range(ââ,â),(ââ,â),vertical asymptotex=0,x=0,andifb>1,b>1,the function is increasing.if0<b<1,0<b<1,the function is decreasing.SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
ifb>1,b>1,the function is increasing. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
if0<b<1,0<b<1,the function is decreasing. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=logb(x+c)f(x)=logb(x+c)shifts the parent functiony=logb(x)y=logb(x)horizontallyleftccunits ifc>0.c>0.rightccunits ifc<0.c<0.SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
leftccunits ifc>0.c>0. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
rightccunits ifc<0.c<0. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=logb(x)+df(x)=logb(x)+dshifts the parent functiony=logb(x)y=logb(x)verticallyupddunits ifd>0.d>0.downddunits ifd<0.d<0.SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
upddunits ifd>0.d>0. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
downddunits ifd<0.d<0. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
For any constanta>0,a>0,the equationf(x)=alogb(x)f(x)=alogb(x)stretches the parent functiony=logb(x)y=logb(x)vertically by a factor ofaaif|a|>1.|a|>1.compresses the parent functiony=logb(x)y=logb(x)vertically by a factor ofaaif|a|<1.|a|<1.SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
stretches the parent functiony=logb(x)y=logb(x)vertically by a factor ofaaif|a|>1.|a|>1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
compresses the parent functiony=logb(x)y=logb(x)vertically by a factor ofaaif|a|<1.|a|<1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When the parent functiony=logb(x)y=logb(x)is multiplied byâ1,â1,the result is a reflection about thex-axis. When the input is multiplied byâ1,â1,the result is a reflection about they-axis.The equationf(x)=âlogb(x)f(x)=âlogb(x)represents a reflection of the parent function about thex-axis.The equationf(x)=lo... | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=âlogb(x)f(x)=âlogb(x)represents a reflection of the parent function about thex-axis. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The equationf(x)=logb(âx)f(x)=logb(âx)represents a reflection of the parent function about they-axis. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
A graphing calculator may be used to approximate solutions to some logarithmic equations SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
All translations of the logarithmic function can be summarized by the general equationf(x)=alogb(x+c)+d.f(x)=alogb(x+c)+d.SeeTable 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Given an equation with the general formf(x)=alogb(x+c)+d,f(x)=alogb(x+c)+d,we can identify the vertical asymptotex=âcx=âcfor the transformation. SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Using the general equationf(x)=alogb(x+c)+d,f(x)=alogb(x+c)+d,we can write the equation of a logarithmic function given its graph. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. SeeExample 3,Example 4, andExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. SeeExample 6,Example 7,andExample 8. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. SeeExample 9,Example 10,Example 11, andExample 12. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. SeeExample 13. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The change-of-base formula is often used to rewrite a logarithm with a base other than 10 andeeas the quotient of natural or common logs. That way a calculator can be used to evaluate. SeeExample 14. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When we are given an exponential equation where the bases arenotexplicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. SeeExample 2,Example 3, andExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can solve exponential equations with basee,e,by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When given an equation of the formlogb(S)=c,logb(S)=c,whereSSis an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equationbc=S,bc=S,and solve for the unknown. SeeExample 9andExample 10. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can also use graphing to solve equations with the formlogb(S)=c.logb(S)=c.We graph both equationsy=logb(S)y=logb(S)andy=cy=con the same coordinate plane and identify the solution as thex-value of the intersecting point. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
When given an equation of the formlogbS=logbT,logbS=logbT,whereSSandTTare algebraic expressions, we can use the one-to-one property of logarithms to solve the equationS=TS=Tfor the unknown. SeeExample 12. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. SeeExample 13. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
The basic exponential function isf(x)=abx.f(x)=abx.Ifb>1,b>1,we have exponential growth; if0<b<1,0<b<1,we have exponential decay. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can also write this formula in terms of continuous growth asA=A0ekx,A=A0ekx,whereA0A0is the starting value. IfA0A0is positive, then we have exponential growth whenk>0k>0and exponential decay whenk<0.k<0.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can find the age,t,t,of an organic artifact by measuring the amount,k,k,of carbon-14 remaining in the artifact and using the formulat=ln(k)â0.000121t=ln(k)â0.000121to solve fort.t.SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Given a substanceâs doubling time or half-time, we can find a function that represents its exponential growth or decay. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use Newtonâs Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Any exponential function with the formy=abxy=abxcan be rewritten as an equivalent exponential function with the formy=A0ekxy=A0ekxwherek=lnb.k=lnb.SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We use the command âExpRegâ on a graphing utility to fit function of the formy=abxy=abxto a set of data points. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We use the command âLnRegâ on a graphing utility to fit a function of the formy=a+bln(x)y=a+bln(x)to a set of data points. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
We use the command âLogisticâ on a graphing utility to fit a function of the formy=c1+aeâbxy=c1+aeâbxto a set of data points. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/6-key-concepts |
f ( x ) = b x , where b > 0 , b â 1 f ( x ) = b x , where b > 0 , b â 1 | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
f ( x ) = a b x , where a > 0 , b > 0 , b â 1 f ( x ) = a b x , where a > 0 , b > 0 , b â 1 | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
A ( t ) = P ( 1 + r n ) n t , where A ( t ) is the account value at time t t is the number of years P is the initial investment, often called the principal r is the annual percentage rate (APR), or nominal rate n is the number of compounding periods in one year A ( t ) = P ( 1 + r n ) n t , where A ( t ) is the account... | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
A ( t ) = a e r t , where A ( t ) = a e r t , where t t is the number of unit time periods of growth a a is the starting amount (in the continuous compounding formula a is replaced with P, the principal) e e is the mathematical constant, e â 2.718282 e â 2.718282 | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
General Form for the Translation of the Parent Function f ( x ) = b x f ( x ) = b x | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
f ( x ) = a b x + c + d f ( x ) = a b x + c + d | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
For x > 0 , b > 0 , b â 1 , x > 0 , b > 0 , b â 1 , y = log b ( x ) y = log b ( x ) if and only if b y = x . b y = x . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
For x > 0 , x > 0 , y = log ( x ) y = log ( x ) if and only if 10 y = x . 10 y = x . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
For x > 0 , x > 0 , y = ln ( x ) y = ln ( x ) if and only if e y = x . e y = x . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
General Form for the Translation of the Parent Logarithmic Function f ( x ) = log b ( x ) f ( x ) = log b ( x ) | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
f ( x ) = a log b ( x + c ) + d f ( x ) = a log b ( x + c ) + d | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
log b ( M N ) = log b ( M ) + log b ( N ) log b ( M N ) = log b ( M ) + log b ( N ) | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
log b ( M N ) = log b M â log b N log b ( M N ) = log b M â log b N | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
log b ( M n ) = n log b M log b ( M n ) = n log b M | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
log b M = log n M log n b n > 0 , n â 1 , b â 1 log b M = log n M log n b n > 0 , n â 1 , b â 1 | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
For any algebraic expressions S S and T T and any positive real number b , b , where b S = b T b S = b T if and only if S = T . S = T . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
For any algebraic expression S and positive real numbers b b and c , c , where b â 1 , b â 1 , log b ( S ) = c log b ( S ) = c if and only if b c = S . b c = S . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
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