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In this lesson, exponential functions with base $e$ and the key features of their graphs will be discussed. Furthermore, logarithmic functions will be introduced and graphed. Finally, the relationship between exponential and logarithmic functions will be explained both algebraically and graphically.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

- Function
- Function notation
- Inverse Function and Finding the Inverse of a Function
- Exponential Function
- Logarithms and Power Property of Logarithms
- Common Logarithm
- Natural Logarithm

Explore

As soon as Zosia entered the room for her Algebra lesson, her teacher asked a question to the class.

Help Zosia find the answer by considering the following questions.

- How do the $y-$values change for one unit increase in $x?$ Is it a constant change?
- Does the table represent a linear relationship?

Discussion

The rate of change for linear functions is constant. For each step in the $x-$direction, the change between $y-$values is the same. Therefore, the *difference* between $y-$values for consecutive $x-$values is the same. Conversely, the rate of change for exponential functions is *not* constant. This means that the differences between $y-$values for consecutive $x-$values are *not* the same.

In the table on the left, the difference between $y-$values for consecutive $x-$values is $2.$ This means that the rate of change is constant. Conversely, in the table on the right, the differences between $y-$values for consecutive $x-$values are not the same. This means that for this table the rate of change is *not* constant.

The table on the left represents a linear function because it shows a constant rate of change. Conversely, the table on the right represents a non-linear function. Notice how the $y-$values, for the table on the right, are doubled for each step in the $x-$direction. To obtain a $y-$value, the previous value is multiplied by $2.$

The table on the left corresponds to a function with a constant increase. In the table on the right, the increase is not constant. However, the ratio between $y-$values for consecutive $x-$values is always the same. This means that the table on the right corresponds to a function with aPop Quiz

Does the table below correspond to an exponential function? If yes, write only the constant multiplier. If not, write only no.

Discussion

Natural base exponential functions are exponential functions whose base, or constant factor, is the number $e$.

$f(x)=ae_{rx} $

Here, $a$ and $r$ are real numbers that are not equal to $0.$ In considering their positive or negative sign, two observations can be made. - If both $a$ and $r$ are positive, the function is an exponential growth function.
- If $a$ is positive and $r$ is negative, the resulting function is instead an exponential decay function.

The graphs of $f(x)=e_{x}$ and $g(x)=e_{-x}$ are shown.

In general, the graph of a natural base exponential function is always a smooth curve.Method

A natural base exponential function with both $a$ and $r$ equal to $1$ will be considered rather than more complex values for the sake of understanding the basics. *expand_more*
*expand_more*
*expand_more*

$y=ae_{rx}Substitutea=1andr=1 y=e_{x} $

To graph this function, three steps must be followed.
1

Identify the Initial Value and Plot the $y-intercept$

The initial value of an exponential function, $y=ab_{x},$ is the number without an exponent, or the value of $a.$ The initial value is also known as the $y-$intercept $(0,a).$ In this example, the value of $a$ is $1,$ and thus the $y-$intercept of the graph of $y=e_{x}$ is $(0,1).$

2

Use the Constant Multiplier to Find More Points

In natural base functions, the constant multiplier — the number with an exponent — is the number $e.$ This means that when $x-$values increase by $1,$ the $y-$values are multiplied by $e.$ With this information, more points can be plotted on the coordinate plane. It is advised to have not less than three points.

3

Draw the Curve

Lastly, the graph can be drawn by connecting the points with a smooth curve.

It is worth remembering that, in general, the graph of a natural base function is always a smooth curve.

Rule

When interest is compounded infinitely many times, it is said to be *continuously* compounded. Let $A$ be the balance of an account that is continuously compounded, $P$ the initial amount, $r$ the interest rate, and $t$ the time. These values are connected by the following formula.

$A=Pe_{rt}$

Keep in mind that, in this formula, the value of $r$ must be written as a decimal and the time $t$ must be in years. Also, the initial amount $P$ is usually called *principal*.

Recall the compound interest formula.

$A=P(1+nr )_{nt} $

In the formula, there is the account's balance $A$, the initial amount $P,$ the time in years $t,$ and the number of times $n$ the interest is compounded per year.
The formula can be rewritten by using the Power of a Power Property.
$A=P(1+nr )_{nt}⇕A=P[(1+nr )_{n}]_{t} $

If the interest rate is $100%,$ or $r=1,$ and the interest is continuously compounded, the formula can be written in terms of $P,$ $t,$ and $e.$ This is because as $n$ goes to infinity, the value of $(1+n1 )_{n}$ approaches $e.$
$A=P[(1+n1 )_{n}]_{t}n→∞ A=Pe_{t} $

The value of the expression $(1+nr )_{n}$, on the other hand, approaches $e_{r}.$ Examine the following interactive graph to better grasp how this is possible.
Using this last approximation, the final form of the formula can be obtained.

$A=P[(1+nr )_{n}]_{t}n→∞ A=Pe_{rt} $

Example

Zosia wants to visit family in Argentina, and while there she hopes to climb Mount Aconcagua! This is going to cost a pretty penny so she needs to increase her savings. Zosia knows that banks do not offer continuously compounded interest, but she is daydreaming about opening an account that does use it.

Recently, she made $$17$ from selling some old stuff at a garage sale. She would deposit this amount into the account. The amount $A$ in the account after $t$ years is calculated by using the following formula.$A=17e_{t} $

a Graph the given function.

b Use the formula to calculate the balance of this account after $1$ year and $9$ months. Suppose that no deposits nor withdrawals are made and round the answer to two decimal places. Check the answer with the graph.

c Use the graph to estimate in how many years the balance of this account would be $$120.$ Suppose that no deposits nor withdrawals are made. Round the answer to the nearest integer.

a

b About $$97.83$

c About $2$ years

a Start by identifying the initial value and plotting the $y-$intercept.

b Keep in mind that $1$ year and $9$ months are $1.75$ years.

c Identify the point on the curve whose second coordinate is $120.$

a The given function is a natural base exponential function whose initial value is $17.$

$A=17e_{t} $

Therefore, the $y-$intercept of the graph occurs at $(0,17).$ Since time cannot be negative and the balance of the account will always be non-negative, only the first quadrant of the coordinate plane will be considered.
The constant multiplier is $e.$ This means that when $x-$values increase by $1,$ the $y-$values are multiplied by $e.$

Finally, the curve can be graphed by connecting these points.

b First, $1$ year and $9$ months must be expressed in years. Recall that $1$ year has $12$ months. This means that $9$ months are $129 ,$ or $0.75,$ years.

$1year and9months↓1129 =1.75years $

Therefore, $t=1.75$ will be substituted into the equation.
$A=17e_{t}$

Substitute

$t=1.75$

$A=17e_{1.75}$

UseCalc

Use a calculator

$A=97.828245…$

RoundDec

Round to $2$ decimal place(s)

$A≈97.83$

The second coordinate of this point is a bit less than $100.$ Therefore, it is reasonable to have a balance of $$97.83$ after $1$ year and $9$ months.

c This time the graph will be used to estimate after how many years the balance would be $$120.$ To do so, the point with the second coordinate $120$ must be spotted on the graph of the function, and its first coordinate needs to be identified.

The first coordinate of the point is *almost* equal to $2.$ Therefore, the balance would be $$120$ after about $2$ years. Zosia is realizing that it might take her quite some time to meet her financial goal, but she is making a valiant effort and certainly on her way!

Example

Zosia finally saved enough money to visit family in Argentina and set out to climb Mount Aconcagua while there! She is as excited as ever.

Here, her knowledge about natural base exponential functions will be extremely useful to her pursuit of reaching the peak. Zosia knows that the atmospheric pressure decreases as the altitude increases, the pressure at sea level is about $1013$ hecto Pascals (hPa), and the pressure $P(x)$ at an altitude of $x$ meters is given by the following formula.$P(x)=1013e_{-0.000128x} $

To avoid unforeseen problems, Zosia realizes that she can share some responsibilities if she recruits a team to solve such issues. Give her a hand and become a valuable member of her team by answering the following questions. If unable to, the team could get altitude sickness! a The altitude of the mountain's peak, called the summit, is $8849$ meters. Calculate the atmospheric pressure at the summit. Round the answer to the nearest integer.

b Draw a graph of the given exponential function.

c Estimate the altitude where the atmospheric pressure is $700$ hecto Pascals. At this altitude, Zosia will rest for a night or two. Round the answer to the nearest thousand meters.

a About $326$ hPa

b

c About $3000$ m

a Substitute $8849$ for $x$ in the given formula.

b Make a table of values.

c Use the graph from Part B.

a To calculate the atmospheric pressure at an altitude of $8849$ meters, this number will be substituted for $x$ into the given function.

$P(x)=1013e_{-0.000128x}$

Substitute

$x=8849$

$P(8849)=1013e_{-0.000128(8849)}$

▼

Evaluate right-hand side

MultNegPos

$(-a)b=-ab$

$P(8849)=1013e_{-1.132672}$

UseCalc

Use a calculator

$P(8849)=326.359490…$

RoundInt

Round to nearest integer

$P(8849)≈326$

b To graph the given function, a table of values will be made first. Only non-negative values will be considered since it is a mountain climb under consideration.

$x$ | $1013e_{-0.000128x}$ | $y=1013e_{-0.000128x}$ |
---|---|---|

$0$ | $1013e_{-0.000128(0)}$ | $1013$ |

$1000$ | $1013e_{-0.000128(1000)}$ | $≈891$ |

$2000$ | $1013e_{-0.000128(2000)}$ | $≈784$ |

$3000$ | $1013e_{-0.000128(3000)}$ | $≈690$ |

$4000$ | $1013e_{-0.000128(4000)}$ | $≈607$ |

$5000$ | $1013e_{-0.000128(5000)}$ | $≈534$ |

$6000$ | $1013e_{-0.000128(6000)}$ | $≈470$ |

$7000$ | $1013e_{-0.000128(7000)}$ | $≈414$ |

$8000$ | $1013e_{-0.000128(8000)}$ | $≈364$ |

$9000$ | $1013e_{-0.000128(9000)}$ | $≈320$ |

$10000$ | $1013e_{-0.000128(10000)}$ | $≈282$ |

The obtained points will now be plotted on a coordinate plane and connected with a smooth curve. Since only non-negative values are considered, the graph just needs to be drawn in the first quadrant.

c To estimate the altitude where the atmospheric pressure is $700$ hecto Pascals, the graph from Part B will be used. The point with the second coordinate $700$ will be spotted and its $x-$coordinate identified.

Tracing a pen or pencil vertically down from the point of the graph that is even, horizontally, with $y-value$ of $700$ leads to the $x-value$ of $3.$ This means that the altitude with an atmospheric pressure of $700$ hecto Pascals is about $3000$ meters. Zosia feels comfortable knowing she will be able to rest for a few nights at this level.

Discussion

Logarithmic functions are functions that involve logarithms.

$f(x)=g_{b}x,b>0andb =1$

The function $f(x)=g_{b}x$ is the parent function of logarithmic functions. Since logarithms are defined for positive numbers, the domain of the function is $x>0$ and its range is all real numbers. If $b$ is less than $1,$ the graph of the function is decreasing over its entire domain. Conversely, if $b$ is greater than $1,$ the graph is increasing over its entire domain.

Rule

A logarithm and a power with the same base undo

each other.

$g_{b}b_{x}=xandb_{log_{b}x}=x$

In particular, the above equations also hold true for common and natural logarithms.

$g10_{x}=xlne_{x}=x andand 10_{logx}=xe_{lnx}=x $

The general equations will be proved one at a time.

$g_{b}b_{x}$

$g_{b}(a_{m})=m⋅g_{b}(a)$

$xg_{b}b$

$g_{b}(b)=1$

$x(1)$

IdPropMult

Identity Property of Multiplication

$x✓$

$g_{b}x=a⇔b_{a}=x $

This will be used to prove the identity.
Therefore, $b$ to the power of $g_{b}x$ is equal to $x.$
$f(x)=b_{x}andg(x)=log_{b}x $

By using the Inverse Property of Logarithms, the composition of these functions results in the identity function.
$f(g(x))=b_{log_{b}x}⇕f(g(x))=x g(f(x))=log_{b}b_{x}⇕g(f(x))=x $

This means that exponential and logarithmic functions are inverse functions. Therefore, the graphs of these functions are each other's reflection across the line $y=x.$ Example

Zosia is getting everything ready for the multi-day hike in Mount Aconcagua. Because her extra battery pack is quite limited, she wants her smartphone and camera to have their batteries fully charged.

The night before the hike, Zosia charges her smartphone and camera. She does not want to leave her devices plugged all night long because it is a waste of energy and could damage the batteries. She read in the user manuals of both devices that the charge in the batteries can be modeled by two logarithmic functions.$Phone’s Batteryp(x)=lnx Camera’s Batteryc(x)=2g(x−0.5) $

Here, $p(x)$ and $c(x)$ are the phone's and camera's charge, respectively, $x$ hours after the devices were plugged in. Zosia wants to graph these two functions on separate coordinate planes. This will help her understand at what time to unplug her devices. Help Zosia graph these functions!
**Phone's Battery:**

**Camera's Battery:**

To graph $p(x)=lnx,$ first graph its inverse function $g(x)=e_{x}$ and then reflect the curve across the line $y=x.$ To graph $c(x)=2g(x−0.5),$ make a table of values. Consider its domain first.

The logarithmic functions will be graphed one at a time.

$p(g(x))=lne_{x}⇕p(g(x))=x g(p(x))=e_{lnx}⇕g(p(x))=x $

Therefore, $p(x)$ and $g(x)$ are inverse functions. This means that the graph of $p(x)$ is a reflection across the line $y=x$ of the graph of $g(x).$ Since $g(x)$ is a natural base exponential function, the graph can be drawn by using its initial value and constant multiplier.
$c(x)=2g(x−0.5)x−0.5>0⇔x>0.5 $

The domain of the function is the set of all real numbers greater than $0.5.$ To make the table, only values in the domain will be considered. $x$ | $2g(x−0.5)$ | $c(x)=2g(x−0.5)$ |
---|---|---|

$1$ | $2g(1−0.5)$ | $≈-0.6$ |

$2$ | $2g(2−0.5)$ | $≈0.35$ |

$3$ | $2g(3−0.5)$ | $≈0.8$ |

$4$ | $2g(4−0.5)$ | $≈1.1$ |

$5$ | $2g(5−0.5)$ | $≈1.3$ |

Example

The expedition was a complete success! What is more, some members of the hiking team felt so inspired by Zosia’s math skills in helping them reach the summit, that they decided to practice some logarithmic functions. They believe this knowledge will also help them in their next expedition to the Amazon Rainforest.

Touched by their passion to learn, Zosia found a few interesting exercises about exponential and logarithmic functions in her online textbook to share with the team.

a Find the inverse function of $f(x)=2g(x−1).$

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b Find the inverse function of $g(x)=51 e_{3x}.$

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a The inverse of $f(x)$ is a common base exponential function.

b The inverse of $g(x)$ is a natural base logarithmic function.

a To find the inverse of the function, the first step is replacing $f(x)$ with $y.$

$f(x)=2g(x−1)↓⏐⏐ y=2g(x−1) $

Now, the $x-$ and $y-$variables need to be switched.
$y=2g(x−1)↓⏐⏐ x=2g(y−1) $

Next, in the obtained equation, $y$ will be isolated.
The definition of a logarithm will be used to express the logarithmic equation as an exponential equation. Since it is a common logarithm, its base is $10.$ $21 x=g(y−1)⇕10_{21x}=y−1 $

Finally, $1$ can be added to both sides of the equation.
$10_{21x}=y−1$

AddEqn

$LHS+1=RHS+1$

$10_{21x}+1=y$

RearrangeEqn

Rearrange equation

$y=10_{21x}+1$

Substitute

$y=f_{-1}(x)$

$f_{-1}(x)=10_{21x}+1$

$x$ | $2g(x−1)$ | $f(x)$ |
---|---|---|

$1.5$ | $2g(1.5−1)$ | $≈-0.6$ |

$5$ | $2g(5−1)$ | $≈1.2$ |

$10$ | $2g(10−1)$ | $≈1.9$ |

$12$ | $2g(12−1)$ | $≈2.1$ |

$18$ | $2g(18−1)$ | $≈2.5$ |

Next, the table that corresponds to $f_{-1}(x)$ will be constructed.

$x$ | $10_{21x}+1$ | $f_{-1}(x)$ |
---|---|---|

$-2$ | $10_{21(-2)}+1$ | $1.1$ |

$-1$ | $10_{21(-1)}+1$ | $≈1.3$ |

$0$ | $10_{21(0)}+1$ | $2$ |

$1$ | $10_{21(1)}+1$ | $≈4.2$ |

$2$ | $10_{21(2)}+1$ | $11$ |

Now the points can be plotted on a coordinate plane and connected with smooth curves. Also, the line $y=x$ will be graphed.

The graphs are each other's reflection across the line $y=x.$ Therefore, $f_{-1}(x)=10_{21x}+1$ is the inverse function of $f(x)=2g(x−1).$

b As in Part A, to find the inverse of the function the first step is replacing $g(x)$ with $y.$

$g(x)=51 e_{3x}⟶y=51 e_{3x} $

Now, the $x-$ and $y-$variables need to be switched.
$y=51 e_{3x}⟶x=51 e_{3y} $

Next, in the obtained equation, $y$ will be isolated.
The definition of a logarithm will be used to express the exponential equation as a logarithmic equation.
$5x=e_{3y}⇔ln5x=3y $

Finally, both sides of the equation can be divided by $3.$
$ln5x=3y$

DivEqn

$LHS/3=RHS/3$

$3ln5x =y$

MoveNumRight

$ba =b1 ⋅a$

$31 ln5x=y$

RearrangeEqn

Rearrange equation

$y=31 ln5x$

Substitute

$y=g_{-1}(x)$

$g_{-1}(x)=31 ln5x$

$x$ | $51 e_{3x}$ | $g(x)$ |
---|---|---|

$-1$ | $51 e_{3(-1)}$ | $≈0.01$ |

$-0.5$ | $51 e_{3(-0.5)}$ | $≈0.04$ |

$0$ | $51 e_{3(0)}$ | $0.2$ |

$0.5$ | $51 e_{3(0.5)}$ | $≈0.9$ |

$1$ | $51 e_{3(1)}$ | $≈4$ |

Next, the table that corresponds to $g_{-1}(x)$ will be constructed.

$x$ | $31 ln5x$ | $g_{-1}(x)$ |
---|---|---|

$0.1$ | $31 ln5(0.5)$ | $≈-0.2$ |

$1$ | $31 ln5(1)$ | $≈0.5$ |

$2$ | $31 ln5(2)$ | $≈0.8$ |

$3$ | $31 ln5(3)$ | $≈0.9$ |

$4$ | $31 ln5(4)$ | $≈1$ |

The obtained points can be plotted on a coordinate plane and connected with smooth curves. Also, the line $y=x$ will be graphed.

It can be seen that the graphs are each other's reflection across the line $y=x.$ Therefore, $g_{-1}(x)=31 ln5x$ is the inverse function of $g(x)=51 e_{3x}.$

a To verify that the obtained function is actually the inverse of the given function, instead of graphing it can be checked that both composite functions $f(f_{-1}(x))$ and $f_{-1}(f(x))$ simplify to $x.$ Start by calculating $f(f_{-1}(x)).$

$f(x)=2g(x−1)⇓f(f_{-1}(x))=2g(f_{-1}(x)−1) $

Now, substitute $f_{-1}(x)=10_{21x}+1$ in the above equation, and simplify as much as possible.
$f(f_{-1}(x))=2g(f_{-1}(x)−1)$

Substitute

$f_{-1}(x)=10_{21x}+1$

$f(f_{-1}(x))=2g(10_{21x}+1−1)$

▼

Simplify right-hand side

SubTerm

Subtract term

$f(f_{-1}(x))=2g(10_{21x})$

$g(10_{m})=m$

$f(f_{-1}(x))=2(21 x)$

AssociativePropMult

Associative Property of Multiplication

$f(f_{-1}(x))=2(21 )x$

DenomMultFracToNumber

$2⋅2a =a$

$f(f_{-1}(x))=1x$

IdPropMult

Identity Property of Multiplication

$f$