Build Your Logarithms Skills with Engaging Lessons and Exercises

Solve the exponential equation $10^{x} = 8.07$ . Express the solution set in terms of natural logarithms or common logarithms.Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

Guadalupe Glass 2022-03-22

Solve the equation for x. $\ln (\ln 3 x) = 0$

London Douglas 2022-03-22

Logistic regression - Transposing formulas
I am trying to understand the math behind Logistic regression. I am confused about transposing one formula to another.
Here is what I have: Our regression formula
$y = b_{0} + b_{1} x$
Our sigmoid function
$p = \frac{1}{1 + e^{- y}}$
Our logistic function
$\ln (\frac{p}{1 - p}) = b_{0} + b_{1} x$
From what I understand, we have to solve for y using the sigmoid function. I did this:
1. $p = \frac{1}{1 + e^{- y}}$
2. $\frac{1}{p} = 1 + e^{- y}$
3. $\frac{1}{p} - 1 = e^{- y}$
4. $\ln (\frac{1}{p} - 1) = \ln (e^{- y})$
5. $\ln (\frac{1}{p} - 1) = - y$
6. $- \ln (\frac{1}{p} - 1) = y$
6 is y for our logistic function, but I don't see how can we get to our logistic function y
$\ln (\frac{p}{1 - p})$
Can anyone help me transpose this or point me in the right direction?
If anyone downvotes me, please explain why in a comment.

resaplaha7dl 2022-03-22

logarithms power equation
I got a home work question to solve the following:
$27 x^{2} < x^{\log_{3} x}$
can any one please explain how to solve this type of equation? I have no idea what to do or what to search for.

Arianna Villegas 2022-03-22

Logarithmic inconsistency when integrating
Consider following integral:
$13 \int \frac{1}{8 x - 4} d x (1)$
By factorizing the denominator and then taking the factor outside the integral sign, it can be rewritten as
$\frac{13}{4} \int \frac{1}{2 x - 1} d x$ (2)
Now (1) and (2) should be equivalent, yet they evaluate into different integrals namely
$13 \int \frac{1}{8 x - 4} d x = \frac{13}{8} \ln | 8 x - 4 | + C$ (1a)
$\frac{13}{4} \int \frac{1}{2 x - 1} d x = \frac{13}{8} \ln | 2 x - 1 | + C$ (2a)
Since $(1) \equiv (2)$ , then (1a) and (2a) should be equivalent as well, which reduces to
$\ln | 8 x - 4 | = \ln | 2 x - 1 |$
which clearly isn't true. What am I missing here?

Anika Boyd 2022-03-22

Logarithmic expression with three terms
So, I have this logarithmic expression
$\log_{5} 8 - \log_{5} 20 - \log_{5} 10$
that I know how to evaluate - the quotient of numbers is the difference of logarithms, so you divide, etc. - but how the heck do you solve one like this with three logs?
I've been able to solve others that only had two logs, but this one confuses me. Help?

Javion Kerr 2022-03-22

logarithm problem - four tuple
How many distinct four tuple (a,b,c,d) of rational numbers are there with
$a \log_{10} 2 + b \log_{10} 3 + c \log_{10} 5 + d \log_{10} 7 = 2005$
Can we proceed like this :
Using $\log a + \log b = \log (a b)$ and $m \log a = {\log a}^{m}$
$\Rightarrow \log_{10} 2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} = 2005$
Please guide how to proceed further..

encaderpwp 2022-03-21

Solve the equation. Find the exact solution if possible; otherwise, use a calculator to approximate to two decimal places.
$3^{3 x - 5} = 7$

jisu61hbke 2022-03-21

Solve the equation $e^{- 2 x + 1} = 13$

Overath097p 2022-03-21

Looking for help understanding the asymptotic expansion of the digamma function
I was recently given an example using this asymptotic expansion of the digamma function where:
$\frac{d}{d x} (\ln Γ (x)) = ψ (x) \sim \ln (x) - \frac{1}{2 x} - \frac{1}{12 x^{2}}$
Here's the example:
$\frac{ψ (\frac{x}{4})}{4} - \frac{ψ (\frac{x}{5} + \frac{1}{2})}{5} - \frac{ψ (\frac{x}{20} + \frac{1}{2})}{20} \sim - \frac{\ln (4)}{4} + \frac{\ln (5)}{5} + \frac{\ln (20)}{20} - \frac{1}{2 x} - \frac{11}{8 x^{2}}$
I'm unclear on the following points:
What happened to each x term?
Why is the first term negative and the rest of the terms positive? Why isn't the signs of the original terms would be preserved?
I would have expected something like this:
$\frac{\ln (\frac{x}{4})}{4} - \frac{\ln (\frac{x}{4})}{4} - \frac{\ln (\frac{x}{4})}{4} - \dots$
How is $\frac{11}{8 x^{2}}$ being determined? Why does $- \frac{1}{12 x^{2}}$ change but $- \frac{1}{2 x}$ stays the same?
Sorry for the elementary questions. The explanation will really help! :-)
Thanks,

peggyleuwpodg 2022-03-21

Logarithms - Get the second solution to a logarithm equation
Here's the original equation:
$\ln (11 x - 10) + {(\ln (11 x - 10))}^{2} = 6$
I've managed to obtain one solution: $x = \frac{e^{2} + 10}{11}$
through those steps:
1. $\ln ((11 x - 10) {(11 x - 10)}^{2}) = 6$
2. $\ln ({(11 x - 10)}^{3}) = 6$
3. ${(11 x - 10)}^{3} = e^{6}$
4. $11 x - 10 = e^{2}$
5. $11 x = e^{2} + 10$
6. $x = \frac{e^{2} + 10}{11}$
The textbook shows a second solution: $x = \frac{10 e^{3} + 1}{11 e^{3}}$ , but how do you get to that result?

Dominique Pace 2022-03-21

Log of negative numbers
I know that log of negative numbers is complex numbers. But I just got over this little proof and wondering what is wrong with this?
$\log (- a) = \frac{2 \times \log (- a)}{2} = \frac{1}{2} \log (a^{2}) = \frac{2}{2} \log (a) = \log (a)$

r1fa8dy5 2022-03-21

Logarithm simplification
Simplify: $\log_{4} (\sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}})$
Can we use the formula to solve this: $\sqrt{a + \sqrt{b}} = \sqrt{\frac{{a + \sqrt{a^{2} - b}}}{2}}$
Therefore first term will become: $\sqrt{\frac{1}{2}}$
$\log_{4}$ can be written as $\frac{1}{2} \log_{2}$
Please guide further..

Tony Mccarthy 2022-03-21

Logarithm rules, which one has priority? $\ln 2 e^{2 x}$
$\ln 2 e^{2 x}$
Here are the two results I came up with:
$2 x (\ln 2 e)$
$2 x (\ln 2 + \ln e)$
$2 x (\ln 2 + 1)$
$2 x \ln 2 + 2 x$
and
$\ln 2 + {\ln e}^{2 x}$
$\ln 2 + 2 x \ln e$
I am sort of leaning towards the first result I got but I am not really sure. Could someone explain whether or not it is correct? I have looked at the log rules but I cannot recall which ones have priority over others.

avalg10o 2022-03-20

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
$\log_{5} (2 x + 3) = \log_{53}$

ashes86047xhz 2022-03-20

Solve the inequality $x^{(\log_{2} x)} + 2 \leq 8$

DeamyKetSate1m 2022-03-19

Baffling identity: $\prod \log_{10} \tan = \sum \log_{10} \tan$
I am quite a baffled now, I am not getting by how it can be written that :
$\log_{10} \tan 40 ° \cdot \log_{10} \tan 41 ° \cdot \log_{10} \tan 42 ° \cdot \log_{10} \tan 43 ° \dots \log_{10} \tan 50 ° ¯ \log_{10} \tan 40 °$
$+ \log_{10} \tan 41 ° + \log_{10} \tan 42 ° + \log_{10} \tan 43 ° + \dots + \log_{10} \tan 50 °$
Is it even valid ? If yes,how ?

Carlo Sawyer 2022-03-18

Any idea how to solve this equation?
$x^{2} \log_{3} x^{2} - (2 x^{2} + 3) \log_{9} (2 x + 3) = 3 \log_{3} \frac{x}{2 x + 3}$

FrokeFofResdts 2022-03-18

Apparently cannot be solved using logarithms
This equation clearly cannot be solved using logarithms.
$3 + x = 2 ({1.01}^{x})$
Now it can be solved using a graphing calculator or a computer and the answer is $x = - 1.0202$ and $x = 568.2993 .$
But is there any way to solve it algebraically/algorithmically?

America Patton 2022-03-18

changing equation to logarithmic form and solving it
$3^{x + 1} = 3000$
How do I solve this? I know we use logarithms but I do not remember how to solve this kind of problem. I am guessing that I need to change the problem into log form. but how?
$\log_{3} (x + 1) = 3 + \log_{10} 3$
what do I do next?

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