Build Your Logarithms Skills with Engaging Lessons and Exercises

Doubt about the domain in logarithmic functions.
According to my book, the logarithmic function
$\log_{a} x = y$
is defined if both x and a are positive and $x \neq 0$ and $a \neq 1$ .
So are these not correct?
$\log_{- 3} 9 = 2$
$\log_{- 2} - 8 = 3$

beefypy 2022-10-16

When is $\frac{\log (a)}{\log (b)}$ an integer?
I've encountered this quite a bit.
If I have $\frac{\log (a)}{\log (b)} = c$ where $b$ is a known positive integer, what can be said about $a$ if $c$ needs to be an integer?

Krish Logan 2022-10-16

Solving a log equation for two variables
Goal is to find both $β$ and $ω$ . I already have the answer here, but I'm confused as to how to get it.
$\log_{6} 250 - \log_{β} 2 = 3 \log_{β} ω$
This is what I did:
$\log_{6} 250 = \log_{β} ω^{3} + \log_{β} 2$
$\log_{6} 250 = \log_{β} 2 ω^{3}$
$\frac{\log 250}{\log 6} = \frac{\log 2 ω^{3}}{\log β}$
And I am stuck here. The answer states that $ω = 5$ and $β = 6$ , which after entering it, is correct, but I don't know how it got to that point.
Looking directly at the last part I ended up with, it seems like you're supposed to equate the tops to one another and bottoms to one another, which would get you the answer, but with another example, that clearly does not work:
$\frac{\log 64}{\log 4} = \frac{\log 27}{\log 3}$
$64 \neq 27, 4 \neq 3$
How would one solve this problem, and is the path I went correct?

Kendrick Finley 2022-10-16

What is the value of $\ln (e^{2 π i})$
I know that
$e^{2 π i} = 1$
so by taking the natural logarithm on both sides
$\ln (e^{2 π i}) = \ln (1) = 0$
however, why isn't this $2 π i$ as expected? Is it beacuse logarithms can only provide real values?

Madilyn Quinn 2022-10-16

Show that if $1 > x > 0$ , then $x - 1 \geq \ln (x) \geq 1 - \frac{1}{x}$
I know how to prove it using the MVT and I can prove it for $x > 1$ but I don't understand how to prove it for $x > 0$

Gisselle Hodges 2022-10-16

When can I assume $Log$ means the natural log?
I have a problem that says:
Express $e^{Log (3 + 4 i)}$ in the form $x + i y$
However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I can interpret ''Log'' as the natural log?

Francis Oliver 2022-10-16

Calculate $\ln 97$ and $\log_{10} 97$ without calculator accurate up to 2 decimal places.
I have rote some value of logs of prime numbers up to $11$
$97$ is a little big.
In case it would have been a multiple of smaller primes, I would have used the trick of logarithm identities .
But I am confused how to do it, or will it be ok to approximate it to 96 or 98.
I also don't know much calculus except differentiation and integration.
I look for a short and simple way.
I have studied maths up to 12th grade.

Mattie Monroe 2022-10-16

Evaluate the limit $lim_{x \to 0} \frac{- z x \ln (z x)}{- x \ln x}$
Title says it all. How can I evaluate the limit provided as:
$lim_{x \to 0^{+}} \frac{- z x \ln (z x)}{- x \ln x}$
I do know that it is equal to $z$ . What I do not know is why.

Pellagra3d 2022-10-16

Using Riemann sums to show that $\sum_{i = 1}^{n} 1 / i = \log n + c + O (1 / n)$
I want to show that there exists a constant $c$ such that:
$\sum_{i = 1}^{n} 1 / i = \log n + c + O (1 / n)$
I am thinking about Riemann sums. Any hints on that?

Bairaxx 2022-10-16

How to find the general sum formula of this logarithmic series $\log 5 + \log 5 + \log 605 + \log 6655 + \dots$

Jaelyn Payne 2022-10-16

Logarithm equations
It has been years since I had to take a math course. I am currently in an advanced Computer Science course and I have been hit with a hw assignment that is bringing back all the math I used to crank out.
Can someone help me understand this question?
Show that $\log_{a} x = c \log_{b} x$ for some constant c (expressed only in terms of the constants a and b).

Amiya Melendez 2022-10-16

Logarithmic problem with 2 variables help
How on earth do I solve? Any help will be much appreciated.
The value of $M$ is given by $M = a \log_{10} S + b$
Note: Seismic moment measure the energy of the earthquake.
Using the following information, determine the values of $a$ and $b$ and hence find the seismic moment $(S)$ of an earthquake which has a magnetic moment $(M)$ of $7.9$
Seismic Moment $(S) = 4.47 \times 10^{25}$ , Magnetic Moment $(M) = 7$
Seismic Moment $(S) = 2 \times 10^{27}$ , Magnetic Moment $(M) = 7.5$

Sariah Mcguire 2022-10-16

What is the sum of recursive logarithms?
I am trying to deduce the complexity of a rather odd algorithm. I have got it down to this form:
$O (n \times (\sqrt{n})^{2} + n \times (\lg \sqrt{n})^{2} + n \times (\lg \lg \sqrt{n})^{2} + . . . + n \times a^{2})$
$= O (n^{2} + n \times (\lg n)^{2} + n \times (\lg \lg n)^{2} + . . . + n \times a^{2})$
Where $a$ is a constant value.
I have a hunch that in the limit to infinity $n^{2}$ dominates. However I do not know how to prove this. Is there a closed form for the sum of recursive logarithms, such as I have in the above formula?

Jaelyn Payne 2022-10-15

Interval of the solutions to $\log_{1 / 2} \log_{2} (\frac{1 + 2 x}{1 + x}) > 0$ is?
I consistently get x>−1 but that doesn't fit the possible solutions I've got.
First step I do is state that $\log_{2} (\frac{1 + 2 x}{1 + x}) < 1$
Then express the $1$ as $\log_{2} 2$ and so on.
What am I doing wrong?

Amiya Melendez 2022-10-15

Solve $2^{x} = 13 mod 3^{4}$
I know $\log 13 = 30 mod 3^{4}$ and $\log 16 = 15 mod 3^{4}$
I've tried subbing $\log 13 / \log 16$ for $2$ but I am not sure what to do next.

tikaj1x 2022-10-15

Prove $\log (x)^{10} < x$ (for $x > 10^{10}$ )
I need to prove that $\log (x)^{10} < x$ for $x > 10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.

Ryder Ferguson 2022-10-15

What is the inverse function of $e^{x} + x$
As the natural $\log (x)$ function is the inverse of the exponential $e^{x}$ and $\log (x + 1)$ is the inverse of $e^{x} - 1$ , what it the inverse of $e^{x} + x$ ?

Nigro6f 2022-10-15

Logarithm and trigonometry
Is $\ln (\sin x - \cos x)$ equal to $\ln (\cos x - \sin x)$ ?
So I did a integral problem but the answer is not same the answer given.
I'm given this question $\int (\frac{2}{1 - \tan x}) d x$
So I got the answer which is $x - \ln (\tan x - 1) + \frac{1}{2} \ln (\sec^{2} x) + c$ Then I simplify it into $x - \ln (\sin x - \cos x) + c$
But the given answer is $x - \ln (\cos x - \sin x) + c$

propappeale00 2022-10-14

How do I evaluate the sum $\sum_{k = 1}^{\infty} (\ln (1 + \frac{1}{k + a}) - \ln (1 + \frac{1}{k + b}))$
where $0 < a < b < 1$ ?
Hints will be appreciated
Thanks

omgespit9q 2022-10-14

Laws of Logarithms
$\log 1050 = \log 73.3 + 0.75 \log m$
I tried this: $\log 1050 = \log 73.3 m^{0.75}$
$1050 / 73.3 = 73.3 (m^{0.75}) / 73.3$
$m^{0.75} = 14.3247$
logm14.3247=0.75
How do you solve for m?

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