Logarithms Definition and 257 Threads

In my working i have, ##\dfrac{\log_{11} x }{\log_{11} 4}= \log_{11} (x+6)## ##\dfrac{\log_{11} x }{0.5781}= \log_{11} (x+6)## ##\log_{11} x = \log_{11} \left[(x+6)\right]^{0.5781}## ##x^{1.729} = x+ 6## ##x^{1.729} -x-6=0## Having ##f(x) = x^{1.729} -x-6## At this point i made use of...

Hi, PF Trigonometric Integrals "The method of substitution is often useful for evaluating trigonometric integrals" (Calculus, R. Adams and Christopher Essex, 7th ed) Integral of cosecant...

Mathologer (https://en.wikipedia.org/wiki/Burkard_Polster) has a nice video using known (but not well-known) geometric motivations of the natural logarithm and the hyperbolic functions... and he makes brief mentions of special relativity I've been using similar motivations to support geometric...

The whole solution is a bit long, which I'll attach but the part I'm stuck at is, assuming everything else above it is correct, is 4 < (log x - 3)(8-log x) Note ; inequalities aren't technically taught yet in the course, so please try to make the solution not go too deep into that. If that...

Part (a) ##s=ab^t## ##\log s= \log a+ t\log b## Expression on the right hand side increases linearly with ##t##Part(b) ##s=120 ×1.15^t## ##\log s = \log 120+t\log 1.15## ##\log s =2.08+0.06t## From graph, y-intercept = ##2.08## ##m=\dfrac{2.45-2.08}{6-0}=0.06## Part (c)...

My attempt that doesn’t work. t=-RC ln(1-I/6.7)

So basically this is how I solved this problem: 1. ##f(x)=\log _{2} x^2 - 1## 2. ##0=\log _{2} x^2 -1 ## 3. ##1= 2\times \log _{2} x## 4. ##\frac{1}{2}= \log _{2} x## 5. ##2^{\frac{1}{2}}=x=\sqrt{2}## So I wrote coordinates to be (##\sqrt{2}##, 0) But apparently, that is not the only solution...

logyx + logxy = 3/2 Solution $$\begin{align*}\log_{ y }{ x } + \log_{ x }{ y } &= \frac{ 3 }{ 2 } \\ \log_{ x }{ y } &= \frac{ \log_{ y }{ y } }{ \log_{ y }{ x } } \\ \log_{ y }{ x } + \frac{ 1 }{ \log_{ y }{ x } } &= \frac{ 3 }{ 2 } \\ \left(\log_{ y }{ x } \right)^ { 2 } + 1 &=...

Hello, I was in class and came up with the question of: is there any physics formula in which a number with units is part of the exponent of said formula, and if there is how do the units behave? Such as for example (x meters)^(y seconds) Thank you in advance.

So in my Calculus book, it asked a question in its Transcendental Functions chapter. It wanted me to express ##3^{\sqrt{2}}## in terms of natural logarithms I have no idea how to solve this. All I know is that ##3^{\sqrt{2}} = e^{\sqrt{2}\ln{3}}## but that's not completely in natural logarithm form.

Is there any way to answer the question without just calculating it using calculator, maybe manipulating the number using logarithm properties? Thanks

In my approach, i made use of change of base; i.e $$x-y=\frac {log_b n}{log_b a} -\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n - log_b n ⋅logba}{log_b a ⋅log_bc}$$ and $$x+y=\frac {log_b n}{log_b a} +\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n + log_b n ⋅logba}{log_b a...

I want to approximate the logarithm of the Binomial coefficient log (n!/ ((n - m)! m!) with the the Stirling approximation log x! ≈ x log x - x I got n log n - m log m - (n - m) log(n - m) but I want (n - m) log (n/(n - m)) + m log (n/m) Can someone help how to transform the first...

This is my work. I don't know whether this is correct or not. What do you think of the problem. Thanks!

<mentor: change title> In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as ##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)## where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term. There is a...

I know that sinhx = 1/2(e^x-e^-x) and that e^2x-1 = e^x(e^x-e^-x) and similar identities but don't know how to get any further. Any hints at where to go with this would be appreciated.

Homework Statement: Suppose f(n) is a function that accepts an integer n as a parameter. When called with n = 1, it executes 2 instructions. When called with a larger value of n, it executes 10n + 30 instructions, two of which are f(n/2). Prove that f(n) executes 10n lg n + 32n − 30...

My Question : Shouldn't differentiating ##-log B## give ##\frac{-\delta B}{B}##? (Note : A, B and Z are variables not constants) By extension for ##Z=A^a \,B^b\, C^c## where ##c## is negative, should ##\frac{\Delta Z}Z=|a|\frac{\Delta A}A+|b|\frac{\Delta B}B-|c|\frac{\Delta C}C##?

Hello! I am reading Schwarz book on QFT and I am at the renormalization group part. I understand the math and the fact that grouping terms perturbatively and requiring them to be 0, when taking the derivatives with respect to a given arbitrary chosen scale, allows perturbative corrections to the...

I was just doing a homework problem that involved logarithms. I noticed that order of operations matters when applying logarithm rules. I'll use a different example from my homework problem to illustrate what I'm talking about. ln(5*2^3) does not equal 3*(ln(5*2)) Apparently you have to do...

Hello I'm having trouble solving for this exponential equation : 16^{x}-(5,4)^{x}-6=0 I used some logarithms properties but can't get anything close to the following solutions here I tried using log base 16 : log16(16^{x})-6=log16((5,4)^{x}) ; then x - xlog16(5,4)=6 ; factorizing x ...

Without using a calculator, prove which of the following is larger: \log_7 10 or \log_{11} 13

Hello all, I have a few small questions regarding logarithms, which I would like to ask your help with. 1) A car loses 50% of it's value every 4 years. How many years does it take for the car to lose 1/3 of it's value ? (I think you need log with basis of 2 here, but not sure) 2) A company...

Homework Statement My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just canceled the relevant n terms and then identified 1/n as the derivative of L however if this were...

I went through an example question that showed me how to solve the question but I'm not sure if I've misunderstood something or if they did a mistake. Question: Derivate y = (1/ax)ax ln(y) = ln( (1/ax)ax ) = ax( ln(1) - ln(ax) ) = -ax ln(ax) (1/y)(dy/dx) = -ax * ax ln(a) - a * ln(ax) dy/dx =...

Homework Statement Evaluate each of the following expressions without using a calculator. 1) log216√8Solve for the unknown value in each of the following equations without using a calculator. 2) 3(x+4)−5(3x)=684 3) 7(42x)=28(4x) Homework Equations Exponent law for multiplication The...

On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...

Homework Statement 2^(t-1) * 6^(t-2) = 20,000 Homework EquationsThe Attempt at a Solution I have no idea how to solve this, although I do understand the basics of logarithms

I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...

Homework Statement A colony of ants will grow by 12% per month. If the colony originally contains 2000 ants how long will it take for the colony to double in size? Answer - 6.12 months Homework Equations A = P(1+r/n)nt The Attempt at a Solution r = 12% = 0.12 n = 12 P = 2000 A = 4000 t = ...

Homework Statement can we put 3log2(x)-4log(y)+log2(5) in one logarithm it try in all the ways but i can't find the solution . Homework Equations loga(b)=logx(b)/logx(a) log(b*a)=log(b)+log(a) The Attempt at a Solution log2(5x^3)-log(y^4) log2(5x^3)-log2(y^4)/log2(10)

Homework Statement [/B] Question: Use the properties of logarithms to rewrite and simplify the logarithmic expression. 1. log_{5}\dfrac{1}{250} Homework Equations Product Property: log_{a}(uv) = log_{a}u + log_{a}v Quotient Property: log_{a}\dfrac{u}{v} = log_{a}u - log_{a}v Power Property...

A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms we define $$\log_b(xy) = \log_b(x) + \log_b(y)$$ Here why is the condition imposed that b>1 and b is not equal to zero and that x and y are positive numbers? Is it something to do with the...

|3^xlog3dxI don't even know where to start. I know that the formula is |u.dv = uv - |v.du u=3^x v=log3

Homework Statement Prove that ##{1/3} < log_{34} 5 < {1/ 2}## Homework Equations ##log_b a = {1/ log_a b}## ##logmn = logm + logn##The Attempt at a Solution ##log_{34} 5 = {1/ log_5 34}## ##= 1/(log_5 17 + log_5 2)## ##=1/(1 + log_5 3 + log_5 2 + something)##...

can somebody help me solving: A*ln(1+A*x)-2*ln(x)-B=0 I thank you in advance

MENTOR note: THread moved from General Math I just can get it This problem has been driving me crazy for a week now If\quad a,b,c\quad \neq \quad 1\\ Also\quad (log_{ b }{ a\cdot }log_{ c }{ a }\quad +\quad log_{ a }{ a })\quad +\quad (log_{ a }{ b\cdot }log_{ c }{ b }\quad +\quad log_{ b...

The question is \log_{x}\left({10}\right)+log(x)=2 where (obviously) I have to find x. I tried changing the base, $\frac{log(10)}{log(x)}+log(x)=2$ $\frac{1}{log(x)}+log(x)=2$ ${log(x)}^{-1}+log(x)=log(100)$ but I could go no further. Whatever I try, I always got a wrong answer. By guessing...

So, there's this problem: A = \frac{1}{6}((\log_{2}\left({3})\right)^3 - (\log_{2}\left({6})\right)^3 - (\log_{2}\left({12})\right)^3 - (\log_{2}\left({24})\right)^3) Find 2^A What I've figured out is that all the logs factorize to 3 + 2 to some power of n...

a>b ⇒ Lim(a–b)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) b>a ⇒ Lim(b–a)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) a∈ℝ b∈ℝ c∈ℝ / c>0

Hi folks, I'm revisiting logs for the first time in a long time through distance education and I was wondering if someone could have a look over a question I've answered and let me know if I've done it correctly or if I'm way off please Find x if Log3(10x – 1) – 2 = 2log3x I instantly divide...

Homework Statement For clarification a have posted the equation below as a picture file.[/B] Homework Equations log(a*b)= loga + logb log(a/b) = loga-logb log(a^n) = nloga[/B] The Attempt at a Solution I don't know how to start. I can't remember the rule for powering the logarithms if there...

I've been teaching myself a little bit of Complex Variables this semester, and I had a question concerning complex integrals. If I understand correctly, then if a function f has an antiderivative F , then the line integral \int_C f(z) dz is path independent and always evaluates to F(z_1)...

Homework Statement Solve the following equation: 2^3+2^3+2^3+2^3=2^x[/B]Homework Equations log(a)^x=x*log(a)[/B]The Attempt at a Solution What i attempted was to log both sides, bring down the exponents, and summarize them. This left me with 12*log(2)=x*log(2). I then divide both sides by...

Homework Statement Proof xlogby=ylogbx Homework Equations logb is a one to one function, that is, if logbx=logby hint take the log in both sides of the equation and use the previous hint The Attempt at a Solution xlogby=ylogbx logby log x = logbx logy logby / logbx=logy/log x...

So, recently I have been learning about logarithms and how to solve exponential equations with the help of logarithms, but I am curious if I can take the log of both sides of an equation like this in order to solve it? If not, then could someone explain why? Thanks.

I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5 My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...

Homework Statement \ln \frac{(x+1)}{(x-1)} \geq 0 Homework Equations \ln \frac {a}{b} = \ln a - \ln b \ln \frac {(x+1)}{(x-1)} = \ln (x+1) - \ln (x-1) The Attempt at a Solution \ln (x+1) \geq \ln (x-1) e^{\ln (x+1)} \geq e^{\ln (x-1)} (x+1) \geq (x-1) According to Wolfram, the...

The problem statement ln(x) = 5 -x Solve for x. The attempt at a solution ln x = 5 - x \\ e^{ln x} = e^{5 - x} \\ e^{ln x} = \frac{e^5}{e^x} \\ x e^x = e^5 Here is the place where I get stuck.