1
vote
4answers
32 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
0
votes
1answer
30 views

Solve for coefficients of $y = A(1 - e^{-x/B})$ given two points

I have the equation $y = A(1 - e^{-x/B})$, and two $(x,y)$ pairs. How can I solve for $A$ and $B$? This should be simple, but I've been banging my head against the algebra for a while to no avail. I ...
3
votes
2answers
139 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
0
votes
1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
3
votes
2answers
60 views

What is the domain of $x^x$

I'm trying to figure out the domain of the function $y=x^x$. When I graph it, it appears to be defined on $[0, \infty)$, but then when I plug in individual negative numbers, for some of them I get ...
1
vote
0answers
70 views

Only one positive solution

When $(a+b)^2=(x-3a)(x-b)e^x$ has only one positive solution, find the relationship between a and b. Here, a and b are constants and satisfy $a>b>0$. Hint: consider the graph of ...
1
vote
1answer
16 views

Derivative of exponential function

1) $f(t) = (\ln 5)^t$ what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong. 2) $f(x) = x^{\Large π^6} + (π^4)^x$ This one I did not attempt in it because I find it confusing little bit.
1
vote
1answer
33 views

A problem with progressive percentage incrementation

I have absolutely no clue if any of the terms I used in the title actually exist or make sense. I'm usually good at math (relatively) but I am facing this problem today that I just cannot solve. John ...
0
votes
3answers
46 views

Exponential problem

$\$10,000$ increases every day by $1\%$. How long until it doubles? I tried doing it by multiplying by $1\%$ for each day and got $10$ days but I don't think that is right. I know there must be an ...
0
votes
1answer
68 views

What is the outdoor temperature? Working included.

Is my working correct in regards to this question? I'm quite stuck on it and I'm not too sure if I am in the right direction. Any advice is appreciated. Thank you. Question: A thermometer that has ...
1
vote
1answer
58 views

Is my working correct? Exponenial decay

Is my working correct? If not, please let me know where I have gone wrong. Thank you for taking the time to check! Question: A thermometer that has been stored indoors where the temperature is 22 ...
3
votes
1answer
113 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
0
votes
1answer
74 views

What is the outdoor temperature? Help please!

Does anyone know how I would go about answering this question? Any feedback is appreciated! A thermometer that has been stored indoors where the temperature is 22 degrees Celsius, is taken outdoors. ...
0
votes
2answers
45 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
1
vote
1answer
46 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
1
vote
1answer
68 views

Solving Exponential Function for termites vs spiders

The populations of termites and spiders in a certain house are growing exponentially. The house contains 120 termites the day you move in. After four days, the house contains 210 termites. Three days ...
1
vote
2answers
25 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
4
votes
3answers
62 views

Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
2
votes
2answers
17 views

Exponential growth precalc population

The population of City A increases by 8% every 10 years. The population of City B triples every 120 years. The two cities had equal populations of 10,000 residents each in the year 2000. In what year ...
0
votes
1answer
33 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
0
votes
1answer
35 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
0
votes
1answer
45 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
vote
0answers
32 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
1
vote
1answer
24 views

Sketching Logs with Quadratic Terms

$\log(x^2+1) = y$ asymptote at $x^2+1 > 0$ and so there is no asymptote $x$ and $y$ intercept at $(0,0)$ How do you know that the function goes both directions, and has a dip in the middle? ...
1
vote
1answer
95 views

(Basic High School Mathematics) Graphing the inverse square law

I did an experiment measuring the intensity of light in relation to the distance away from a source. How would I graph the avg intensity over 1/distance squared? Note that T1 = trial 1 etc.. It's ...
2
votes
7answers
129 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
0
votes
1answer
35 views

Just one step away. Exponential formula

The data points are. I have worked out that. y =.032(2.5)^x is correct where my table is x = 0, y = .032 x = 1, y = .08 x = 2, y = .2 How do I get my formula to reflect y=-3, x =.0320 ...
0
votes
2answers
12 views

Calculating how many iterations to make a list that doubles each iteration, n elements long

If I have a list of numbers starting with four numbers, the list doubles in size after each iteration, how would I calculate how take to have a list of exactly n elements long? Thanks
0
votes
2answers
35 views

Proportionality to find spent years for price drop

Well, the title's kinda messy, but this is a concrete example of what I'm trying to find out: Lets say there is a price of 40.000 USD, if the price drops at half, how many years does it take for the ...
1
vote
1answer
47 views

If$ a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$ then $a=b$?

If $$a+b-1=1+\frac{ln(2^a-1)}{ln4}+\frac{ln(2^b-1)}{ln4}$$ where $a,b>0$ are real numbers and ln is $log_e$, then is a=b?
4
votes
1answer
98 views

Proof of $e^{\ln(x)\ln(2)}$, which natural logarithm do I bring down?

I'm currently stumped with the proof for the following problem: $$F(x) = 2^{\ln(x)}$$ $$\Rightarrow F(x) = y$$ $$y = 2^{\ln(x)}$$ $$\ln(y) = \ln(2^{\ln(x)})$$ $$\ln(y) = \ln(x)\cdot\ln(2)$$ $$y = ...
1
vote
2answers
53 views

How can I solve the equation…

$$\log\left(\frac{\pi_i}{1-\pi_i}\right)=\sum_{k=0}^K x_{ik}\beta_k\qquad i=1,2,\dots,N$$ How can I make the equation above the one below by taking "$e$" to both sides. Note that after taking $e$ ...
0
votes
1answer
40 views

How to solve this equation with linear n as well as polynomial n?

I am banging my head against the wall, but somehow I can't find a closed form solution to this equation in n: $$229,244 + 58,044 \cdot n = 130,000 * 1.78^n$$ Obviously, if there was no $n$ ...
0
votes
1answer
88 views

Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
0
votes
1answer
2k views

Compound interest formula and continuously compounded interest formula derivation

My textbook gives the formula for compound interest as: $A\left( t\right) =P\left( 1+\dfrac {r}{n}\right) ^{nt}$ Where: P = The principal, r=the annual rate of interest, n= the frequency of ...
-1
votes
1answer
57 views

How come is this equals to 1?? [closed]

How is this equal to $1$.. $e^{-i}(\frac{2·\pi·k·(-N)}{N})=1$ ??? http://cnx.org/content/m12024/latest/#Boncelet
0
votes
4answers
45 views

Exponential Growth

A colony if bacteria in a petri dish grows exponentially. At noon, there 1000 bacteria cells in the dish. At 8 pm, there 3000 cells in the dish. How long does it take the bacteria to double its ...
2
votes
3answers
308 views

Solve $2^{x}=x^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating x. I first tried taking the natural log: $x\ln \left( 2\right) =2\ln \left( x\right) $ $\dfrac {\ln \left( ...
0
votes
1answer
68 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
0
votes
3answers
38 views

Why do I receive the wrong answer when I try to solve this exponential equation?

So I have the equation: $25^{x}=5^{x}+6$ My reasoning is if you make everything to the base 5: $\left( 5^{2}\right) ^{x}=5^{x}+5^{\log _{5}6}$ Given the bases are the same we can do: $2x=x+\log ...
5
votes
2answers
206 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
1
vote
1answer
35 views

Help needed clearing up a textbook explanation of logarithms

A passage in my textbook has me confused, first it states this: $ \log _{a}\left( x\right) .\log _{b}\left( a\right) =\log _{b}\left( a^{\log _{a}\left( x\right) }\right) =\log _{b}\left( x\right) $ ...
1
vote
8answers
331 views

Given $2^{x}=129$, why is it that I can use the natural logarithm to find $x$?

I've looked at an example in my textbook, it is: $2^{x}=129$ $\ln \left( 2^{x}\right) =\ln \left( 129\right) $ $x\ln \left( 2\right) =\ln \left( 129\right) $ $ x=\dfrac {\ln \left( 129\right) ...
1
vote
2answers
107 views

Confusion regarding the Logarithmic function change of base formula

My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated. It starts with: $b^{x ...
1
vote
1answer
100 views

How is the gradient of exponential functions with different bases (in the included table) worked out?

I am studying exponential functions at the moment, and this table was presented in my textbook to show that for exponential functions with increasing 'bases' the gradient of the function increases. ...
1
vote
1answer
79 views

Practical significance of $e$ [duplicate]

We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't ...
3
votes
2answers
137 views

Why is it important to define that a logarithm and exponential function is one-to-one?

I'm currently studying the properties of logarithm in an open source pre-calculus textbook that can be found here (Page 438). Before the text goes on to the Algebraic properties of exponential and ...
0
votes
2answers
41 views

Exponent multiplication error

$\left( x^{2/3}\right) ^{3/2}=x^{\dfrac {2}{3}\times \dfrac {3}{2}}=x^{1}$ Given this why is it that if I substitute $x=-1$ I get 1?: $\left( \left( \sqrt [3] {-1}\right) ^{2}\right) ...
2
votes
1answer
78 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$