1
vote
3answers
52 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
votes
2answers
22 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
vote
2answers
18 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
0
votes
1answer
29 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
39 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
15 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
30 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
20 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? ...
0
votes
2answers
65 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
19 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
1
vote
3answers
27 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
1
vote
1answer
28 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
0
votes
3answers
48 views

“Linearize” an exponential-looking graph with log function

This may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize" ...
0
votes
1answer
24 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
0
votes
1answer
37 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
1
vote
2answers
41 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
1
vote
3answers
66 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
17
votes
2answers
478 views

Is this function a constant?

I am a french guest and I hope that my english isn't too bad... So here is my issue : I consider an entire function $f$ which satisfies the following property for all complex number $z\in \mathbb{C}$ ...
0
votes
1answer
151 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
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?
5
votes
2answers
212 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
4
votes
1answer
85 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
50 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 $e$ ...
1
vote
6answers
73 views

Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
3
votes
2answers
74 views

Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
0
votes
1answer
38 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
27 views

Calculating an exponentially increasing vector of points in a test and measure system

My application is setting and measuring current and voltage in a physical system with a software algorithm. Given these parameters: min, ...
2
votes
3answers
303 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
59 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
35 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
180 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 ...
0
votes
1answer
50 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
1
vote
1answer
30 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
313 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
95 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
2answers
574 views

What is the difference between logarithmic decay vs exponential decay?

I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.
3
votes
2answers
108 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 ...
2
votes
2answers
51 views

Changing an exponential function to logarithmic

I have a question stating that $P=75e^{-0.005t}$ and they want to get t by itself. I used the example $y=2^x = x=log_2(y)$ To find that $-0.005t = 75ln(P)$ So $t=\frac{75ln(P)}{-0.005}$ However ...
0
votes
2answers
34 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
2
votes
2answers
94 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
5
votes
1answer
591 views

where do exponential and logarithmic functions intersect?

If $0<a<1$, then the graphs of $y=a^x$ and $y=\log_a(x)$ intersect at some point $(t(a),t(a))$. Does this function $t(a)$ have any nice expression? How much do we know about this function, ...
0
votes
1answer
28 views

Exponential growth/reduction of Laser Intensity question?

Hi i have a pretty simple question but I am not quite sure on how to solve/approach it. THe question: "The Intensity of a laserbeam declines with the penetrationdepth into matter exponentially. At ...
0
votes
1answer
32 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
0
votes
1answer
57 views

Why $\log(-2-y)$ is equivalent to $\log(2+y)$ for restricted values?

I was looking through the step-by-step solution given by Wolfram|Alpha to a problem, and at the last step it says that ... $= -\log(-2-y(t)) + \mathrm{constant}$ Which is equivalent for restricted t ...
1
vote
1answer
54 views

Why is this limit $\frac{e^x}{x^{x-1}}$coming out wrong?

Attempting to answer this question, I thought to evaluate the limit by taking the logarithm and then using L'Hopital's rule: $$\begin{align} L&=\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}\\ ...
0
votes
2answers
94 views

How to go from a sum to a product and a product to a sum?

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
0
votes
1answer
72 views

Question about natural logarithm in the exponent of the e-function

I wonder which rule dictates that e^(-2x+ln(c)) is equal to e^(-2x) * c I know that the logarithm naturalis is the "reverse-function" of the e-function but why isn't it e^(-2x) + c instead?
1
vote
1answer
104 views

How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$

Solve the following equation for $x$ : $ \quad3^{2x+1}-10\cdot 3^x+3=0 $ I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how ...
2
votes
1answer
235 views

Exponential practice exam question

Okay bear with me, this is one of those cumulative questions The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula: $$x = D{e^{ - {1 \over ...
2
votes
3answers
109 views

Show that $x=2\ln(3x-2)$ can be written as $x=\frac{1}{3}(e^{x/2}+2)$

Show that $x=2\ln(3x-2)$ can be written as $x=\dfrac{1}{3}(e^{x/2}+2)$. Is there a rule for this?