Build Your Logarithms Skills with Engaging Lessons and Exercises

How do I solve this equation involving a logarithm?
$x \log (x) = 100$

Kennedy Ford 2022-04-15

Is $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots = \ln 2$ true?

Breanna Fisher 2022-04-15

Is $lim_{x \to \infty} \frac{\ln {x}}{x} = lim_{x \to \infty} \frac{\frac{1}{x}}{1}$
In my notes its given $lim_{x \to \infty} \frac{\ln {x}}{x} = lim_{x \to \infty} \frac{\frac{1}{x}}{1}$
Is that correct? How do I get that?
I think another example is also related
$x^{\frac{1}{x}} = e^{\frac{\ln {x}}{x}}$

Devin Dougherty 2022-04-14

Lambert W / Product log function?
I would like to solve this equation:
$n \cdot 2^{n} = 15000$
And according to WolframAlpha
$n = \frac{W (15000 \log (2))}{\log (2)}, where \log is ln$
Which shows that I need to use the product log function W which I tried looking up on wikipedia. I don't need the complex numbers, just real numbers.
Additionally, are there ways of solving the original equation without the W functions?
Can someone explain the rules and how to do this please?
I would eventually like to implement a way to find n programmaticly (if possible-in python).

Brielle James 2022-04-14

Differentiate $y = \sqrt{\frac{1 + 2 x}{1 - 2 x}}$ logarithmically
$y = \sqrt{\frac{1 + 2 x}{1 - 2 x}}$
$\ln y = \frac{1}{2} \ln (1 + 2 x) - \frac{1}{2} \ln (1 - 2 x)$
$1 / y d y / d x = 1 / 2 \times 2 / (1 + 2 x) - 1 / 2 \times - 2 / (1 - 2 x)$
$1 / y d y / d x = 1 / (1 + 2 x) + 1 / (1 - 2 x)$
$d y / d x = (1 - 2 x) + (1 + 2 x) / (1 + 2 x) (1 - 2 x) \times y$
$d y / d x = 2 {(1 + 2 x)}^{1 / 2} / (1 + 2 x) (1 - 2 x) {(1 - 2 x)}^{1 / 2}$
$d y / d x = 2 / {(1 + 2 x)}^{1 / 2} {(1 - 2 x)}^{3 / 2}$
However the answer is:
$d y / d x = 2 / (1 - 2 x) \sqrt{(1 - 4 x^{2})}$
Where have I went wrong?

Axel Stout 2022-04-13

Manipulating log function
I'm trying to find the value of a constant for $y (0) = 0$ in the following differential equation.
$2 \ln (2 x + 3 y - 1) - \frac{2 x + 3 y}{2} = 2 x + 3 y + k$
Of course when plugging in the values, I get $2 \ln (- 1) = k$ which errors. When entering this into Wolfram Alpha, they suggest rearranging the equation from this format,
$2 \ln (x - 1) - \frac{x}{2} + constant$
to
$- \frac{x}{2} + 2 \ln (1 - x) + \frac{1}{2} + constant$
"Which is the equivalent for restricted x values", which indeed I have. This would leave me with a positive $\ln (1)$ , which would solve my problem... but...
My question, how does this manipulation work? I've never seen this before. I don't understand how they've made that leap (or if it's even accurate).
Can anyone educate me?

Emmy Decker 2022-04-13

Is this log form simple enough?
$\frac{3}{\ln 2 - 12}$
Is this form simplified enough?
There is a number '12' below the fraction line, do i need to transform the log more to make it simpler?
I wrote that in a college math exam

Frances Simon 2022-04-13

Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution,when appropriate.
$5^{x} + 125 (5^{- x}) = 30$
we have to find the exact solution.
$5^{x} + \frac{125}{5 x} = 30$
$5^{x} + \frac{125}{5^{x}} = 30$
let $5^{x} = t$

Milton Shepard 2022-04-12

For what number $n$ does the equation $\log_{n} x =^{n} x$ have exactly one solution?

Colten Welch 2022-04-12

Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution,when appropriate.
$2^{- x^{2}} = 5$

gabolzm6d 2022-04-12

Find the equation of the tangent to the curve $y = 2^{x} + 2^{- x}$ at the point $(2, 4 \frac{1}{4})$
$y = 2^{x} + 2^{- x}$
$\ln y = x \ln 2 - x \ln 2$
$\ln y = 0$
$\frac{1}{y} \frac{d y}{d x} = 0$
$\frac{d y}{d x} = 0$
I've checked the answer and I've got the differential wrong, What am I doing wrong? I assume it has something to do with the expression with the negative exponent? Am I not allowed to prefix the natural expression with "minus" x? Could someone explain please, thank you!

Ormezzani6cuu 2022-04-12

Find intersection of linear and logarithmic lines
I have equations for two lines, one of which is linear and the other is logarithmic, ie:
$y = m_{1} x + c_{1}$
$y = m_{2} \cdot \ln (x) + c_{2}$
..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:
$m_{1} \cdot x + c_{1} = m_{2} \cdot \ln (x) + c_{2}$
..for x, but apart from shuffling the constants around I'm not sure how to do this
Is there a general solution to this problem?
Thanks

Alessandra Carrillo 2022-04-11

Use the same-derivative argument, as was done to prove the Productand Power Rules for logarithms, to prove the Quotient Ruleproperty.

Jayleen Aguirre 2022-04-11

Use the properties of logarithms to write each expression as a single term.
a. $\ln [(x + 3) - \ln (x - 1)]$
b. $\frac{1}{2} \log_{7} (2 x - 3) + \log_{7} x - \log_{7} q$

Ean Hughes 2022-04-11

Use the properties of logarithms to solve the following equation for x.
$2 \ln (x + 2) - \ln x = \ln 8$

Leonard Montes 2022-04-11

use the properties of logarithms to rewrite the expression as 3 separate logarithms : $\log_{3} \sqrt[4]{\frac{5}{2}}$

poznateqojh 2022-04-11

Use the properties oflogarithms to simplify the expression.
$3^{x + 1} = 27$

Jazmin Strong 2022-04-11

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places $\log_{π} 400$ .

Adison Palmer 2022-04-11

Unitary matrix in trace and log function
I am trying to do a unitary transformation $U$ on square matrix $A$ which is embedded inside a trace and natural log function, and the following property is supposed to hold:
$t r (\ln (A)) = t r (\ln (U A U^{∔}))$
What property of the $t r$ and $\ln$ functions would allow us to do that?

Jaylyn Villarreal 2022-04-11

Trying to convert a nasty logarithm into an exponential
I have the following equation that I must express in terms of r:
$Δ V = \frac{λ}{2 π ϵ_{0}} \ln (\frac{r}{R})$
This is a pretty tough one. I am not sure how to get the r out of the logarithm. I have tried to express the logarithm as $\ln (r) - \ln (R)$ , , but I am not sure where to go from there, or where the coefficient should go. I know that $Δ V$ goes up in the exponent...

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