Build Your Logarithms Skills with Engaging Lessons and Exercises

Denesting Logarithmic expressions
$\log_{7} (\log_{2} (3)) + \log_{7} (\log_{5} (6)) + \log_{7} (\log_{11} (\frac{1}{2})) = \log_{7} (- 1) + \log_{7} (\log_{5} (3)) + \log_{7} (\log_{11} (6))$
This can only be simplified by using the sum to product rule and noticing that of the n! possible permutations that $\frac{1}{2}$ is a power of 2. Because N is small and the relationship between 2 to $\frac{1}{2}$ is simple this can be solved trivially. How does one go about simplifying much more complex logarithmic identities in polynomial time?

Reuben Brennan 2022-03-25

Convergence of series $\sum_{n = 1}^{\infty} \ln (\frac{2 n + 7}{2 n + 1})$
I have the series
$\sum_{n = 1}^{\infty} \ln (\frac{2 n + 7}{2 n + 1})$
I'm trying to find if the sequence converges and if so, find its sum.
I have done the ratio and root test but It seems it is inconclusive.
How can I find if the series converges?

Harley Ayers 2022-03-25

Condensing logarithms
I got to this: $2 \log_{10} x^{\frac{1}{2}} + 3 \log_{10} x^{\frac{1}{3}}$
Now, usually you bring the exponent the the front and that would yield:
$\frac{1}{2} (2) \log_{10} x + \frac{1}{3} (3) \log_{10} x = \log_{10} x + \log_{10} x = 2 \log_{10} x$
And that's it?

burubukuamaw 2022-03-25

Condense using properties of logarithms: $\log 13 + \frac{13}{\log (x - 9)} - 3 \log (x + 9)$

navantegipowh 2022-03-25

Computing: $lim_{x \to 0} \frac{\log (1 + x)}{x^{2}} - \frac{1}{x}$
Could the following limit be computed without L'Hopital and Taylor? Thanks.

parksinta8rkv 2022-03-25

How can I calculate $lim_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$ . Where a and b are two integers.

juctommaccedo662f 2022-03-24

Hey guys so i'm trying to turn this equation into it's sum/difference logarithm. However, the part that messes me up is turning the bottom of the fraction
${\log (\frac{x^{2} + 2 x + 1}{x^{2} - 3 x + 2})}^{2} .$
I think it will turn into this:
$4 \log (x + 1) - 2 (\log (x - 1) + \log (x - 2)),$
but I don't want it to be ${(x - 1)}^{2}$ times ${(x - 2)}^{2}$ . How do I solve it so it's ${((x - 1) (x - 2))}^{2}$ ?

diartrosiwk8o 2022-03-24

Given, $\log_{10} 2 = 0.30103$ ,then find the number of digits in $2^{56}$

rijstmeel7d4t 2022-03-24

Logarithms equation, litteral
I have problems with the following logarimthic equation:
$\log_{a} (\frac{x + \sqrt{x^{2} + 5}}{5}) = b$
How can I compute $\log_{a} (x - \sqrt{x^{2} - 5})$ in terms of b?

Petrolovujhm 2022-03-24

Logarithmic equations and their related exponential equations
I am learning about logarithms and I'd like some examples of the following:
Examples where the logarithmic value is a different positive integer
Examples where the logarithmic value is a different negative integer
Examples where the logarithmic value is a different non-integer fractional value

nastupnat0hh 2022-03-24

Limit $lim_{x \to \infty} x - x^{2} \ln (1 + \frac{1}{x})$

ropowiec2gkc 2022-03-24

Let $k \in N$ . Prove that
$0 < \frac{1}{k} - \ln (1 + \frac{1}{k}) < \frac{1}{2 k^{2}}$

limes14514qqmn 2022-03-24

Given $1 < a < b < c$ prove
$\log_{a} \log_{a} b + \log_{b} \log_{b} c + \log_{c} \log_{c} a > 0 .$

kembdumatxf 2022-03-24

Given binary tree with N nodes and height h.Prove $[\log_{2} (N)] \leq h$ .

Addison Fuller 2022-03-23

Solve the equation
$22 \log (92 x + 40.66) = 38.9$
The equation
$22 \log (92 x + 40.66) = 38.9$
steps so far
$\log (92 x + 40.66) = \frac{38.9}{22}$
to eliminate log, do I have to apply the opposite of log? Not sure what that is.

Kiara Haas 2022-03-23

Maclaurin series for a function
Provided I have the function
$f (x) = {(1 + x)}^{\frac{1}{x}},$
and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as this seems hard..). I know that
${(1 + x)}^{\frac{1}{x}} = e^{\ln \frac{(1 + x)}{x}},$
and the Maclaurin series for $e^{x}$ is easy to prove, so I think it's a good direction..

Landen Barber 2022-03-23

logarithm of a sum or addition
I search a general rule for calculating the logarithm of a sum or addition. I know that
$\ln (a + b) = \ln (a (1 + \frac{b}{a})) = \ln (a) + \ln (1 + \frac{b}{a})$
but when the sum implies more terms, how to generalize its calculation/computation? For example when
$\ln (a + b + c)$
or when
$\ln (\sum_{i} a_{i})$
like when we have to compute the denominator of a Bayes formula using log-likelihoods? Thanks for your incoming help.

Anika Boyd 2022-03-23

Limit with ' sequence and inverse logintegral
I found formula below
$lim_{n \to \infty} \frac{\lim^{- 1} (n)}{p_{n}} = 1$
where $\lim^{- 1} (n)$ is inverse logintegral function and $p_{n}$ is ' number sequence.
Can anyone prove this formula?

tibukooinm 2022-03-23

Limit of logarithms without l'Hospital
This is my first post so I hope you forgive any formatting mistakes. This is a task out of a training exam, I may add that we have not yet introduced l'Hospital or derivatives. We have to determine the following limit:
$lim_{n \to \infty} \frac{\ln (2 n^{n}) - \ln (\sqrt{n})}{\ln (3 n^{2 n}) + \ln (\sqrt{n})}$
I am stuck on this for quite some time now, I tried to apply the ln sum rules, but i can not find a way to solve this one. I'm thankful for advice.

monkeyman130yb 2022-03-22

Solve the exponential equation
$5^{2 x + 1} + 5^{x} - 4 = 0 .$

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