Build Your Logarithms Skills with Engaging Lessons and Exercises

Growth rate of logarithmic function?
Just curious about the growth rate of the logarithmic function:
Does there exist a real number $n$ such that $l i m_{x \to \infty} \frac{(l n (x))^{n}}{x}$ diverges (does not converge to $0$ )?
Thanks in advance!

Deja Bradshaw 2022-10-12

rules for evaluating powers of logarithms
What rules are we using to show that $3^{- s} = \frac{1}{2}$ if $s = \frac{\log 2}{\log 3}$
I cannot understand how you can raise a number to a logarithm divided by a logarithm

link223mh 2022-10-12

Solving a logarithmic equation $\log_{2} (2^{x} - 1) + x = \log_{4} (144)$

Diego Barr 2022-10-12

The same derivative for very different functions
Just a curiosity and request for a comment on this.
$\frac{d}{d x} (- 2 \log (u x - 2)) = - \frac{2 (x u^{'})}{u x - 2}$
$\frac{d}{d x} (- 2 \log (2 - u x)) = - \frac{2 (x u^{'})}{u x - 2}$
What can be said about such results? Clearly there is nothing similar between $u x - 2$ and $2 - u x$ , yet the equations are both the same. Checked with Mathematica and by simple manual computation. Maybe the differentiated functions can be derived one from another?

Chloe Arnold 2022-10-12

Logarithm proof fallacious or not?
$e^{- x} = e^{1 / x}$ Taking the natural logarithm of both sides
$\ln (e^{- x}) = \ln (e^{1 / x})$
$- x = 1 / x$
$- x^{2} = 1$
$x^{2} = - 1$
$x = i$
I know I am doing something wrong here. Also can someone please explain why
$- \ln (x) = \ln (1 / x)$
Thank you.

Eliza Gregory 2022-10-12

Rewrite logarithmic expression
I have this logarithmic expression
$2 \log_{b} 6 + (1 / 2) \log_{b} 25 - \log_{b} 30$
and have to rewrite it as logb of one number. I just don't understand how to do this. help please.

Jaylyn Horne 2022-10-12

Find:
$\log_{2} x + \log_{2} 2 x = 4$

faois3nh 2022-10-11

Simple Logarithmic question.
I was just wondering if i can do this.
Q. Solve $\log_{9} 24 = x$
$⟹ 9^{x} = 24$
$⟹ 3^{2 x} = 2^{3} 3$
$⟹ \log_{3} (3^{2 x}) = \log_{3} (2^{3} 3)$
$⟹ 2 x = 2 (3)^{1 / 3}$
$⟹ x = 3^{1 / 3}$
Is this actually correct or did i break some kind of log rule here? My answer appears to be out by 0.0038 compared to the book is that cause they used a calculator or is it just a fluke that my answer is so close?

Trace Glass 2022-10-11

How do I solve $3^{\ln 2} \times x^{\ln x + \ln 6 + 1} = \frac{3 e^{2}}{4}$
I've been going at this question for 2 hours, my teacher wants us to solve for x without a graphing calculator.
$3^{\ln 2} \times x^{\ln x + \ln 6 + 1} = \frac{3 e^{2}}{4}$

sorrowandsongto 2022-10-11

What is the Solution? $x \geq 8 \lg x$
I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.

Nikolai Decker 2022-10-11

Logarithm formula proof
Prove:
$x^{\log (y)} = y^{\log (x)}$
I have been trying this for the past 1 hour, still cant prove it.
I started with
$\log_{b} (y) = m$
$\log_{b} (x) = n$
To show:
$x^{m} = y^{n}$
How do i proceed?

Deon Moran 2022-10-11

Integral of $\ln (3 x) / x$
I believe this should be a simple problem but I don't have an answer key to confirm if this is right, and some of the similar questions I can find online seem to be giving more complicated solutions.
The problem is:
$\int \frac{\ln (3 x)}{x} d x$
And this is my solution, based on the fact that $\int \ln (a x) d x = \frac{1}{x}$
$u = \ln (3 x)$
$⟹ \frac{d u}{d x} = \frac{1}{x}$
$⟹ d u = \frac{1}{x} d x$
So:
$\int \frac{\ln (3 x)}{x} d x = \int u d u$
$⟹ \frac{u^{2}}{2} + C$
$⟹ \frac{(\ln (3 x))^{2}}{2} + C$
Is that right?

Cale Terrell 2022-10-11

Logarithms involving decimals
I am a student wondering how would I put this correctly into a calculator.
I have 1,05 and 1,216
1,05^n=1,216
How would I calculate n without just multiplying 1,05 against itself until I hit the desired result.I know that n=4 but how would I calculate this easier?
All tips and help is appreciated.
Thank you in advance!

Maribel Vang 2022-10-11

I have issue in calculating log values
How is
$l o g_{4} 2 = \frac{1}{2}$
?
Any formula to how we calculate this?
I know i am confused when base is larger digit than log value term.

Buszmenan 2022-10-11

Solve simultaneous logarithmic equations with different bases?
How do I solve these simultaneous equations?
$2 l o g_{x} y + 2 l o g_{y} x = 5$
$x y = 8$
I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do
$x = 8 / y$
and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?

Janiah Parks 2022-10-10

Logarithmic Equation: Solve for x
$\log_{3 x} 81 = 2$
How would I go about solving this? This is what I tried:
$\log_{3 x} 81 = 2$
$\frac{\log 81}{\log 3 + \log x} = 2$
Where do I go from here?
If I isolate $\log x$ on one side, how do I get rid of the log?

st3he1d0t 2022-10-09

How do I solve for $t$ in this equation?
I know I'm supposed to use $\ln ()$ to work it out, but I can't remember how it's done. Can anyone help?
The equation is
$40 e^{- t / 5} = 20$

dripcima24 2022-10-09

Logarithmic equation help
$\log_{5} (x + 3) + \log_{5} (3 x - 5) = \log_{25} (9 x^{2})$
I have the answer: ${\frac{\sqrt{181} - 1}{6}}$ (only answer that falls in the domain)
i understand how to work with the left side but can't figure out how to get the RH side to base 5.

Bruce Sherman 2022-10-09

Why does $lim_{x \to \infty} x - x^{{\frac{1}{x}}^{\frac{1}{x}}} - \log^{2} x = 0 ?$
Moreover, why is
$x - x^{{\frac{1}{x}}^{\frac{1}{x}}} \approx \log^{2} x ?$

samuelaplc 2022-10-09

Bounding a logarithmic relation
If I have the following relation $T (n) \leq a n ⌈ \lg (n) ⌉ - a n + 2 b n + n$ , is it possible to bound $T (n)$ such that it is in the form $T (n) \leq a n \lg (n) + b n$ for some constants $a, b \geq 0$ ?

Finding logarithms questions and answers online is always pure luck, especially if these are provided with detailed answers to questions. This is exactly what you can find below regardless of your practical application, college course, or complexity. These logarithms questions are related to higher education courses, which is why they will help you find the answers to what you currently have. When you deal with logarithmic equations you are approaching an exponential equation where the variable appears in an exponent as it’s in (log b (bx) = x. It also shows that the logarithmic graph equation can be represented as well.

Algebra II

Learn How to Graph Logarithmic Equations