Build Your Logarithms Skills with Engaging Lessons and Exercises

Use the base-change formula to find the logarithm to fourdecimal places.
$\log_{5} (2.56)$

John Landry 2022-07-27

Use properties of logarithms to find exactly value.
$5^{\log_{5} 6 + \log_{5} 7}$

Luz Stokes 2022-07-27

Solve for x:
$5^{3 x + 24} = 1$

ganolrifv9 2022-07-25

What is the logarithm of 232 feet and 4 inches ?

Awainaideannagi 2022-07-23

Logarithm Fraction Contest Math Question
The question is as follows:
If $\frac{\log_{b} a}{\log_{c} a} = \frac{19}{99}$ then $\frac{b}{c} = c^{k}$ . Compute $k$

Marcelo Mullins 2022-07-23

Derive an equation for derivative of ln x
$\frac{d}{d x} e^{x} = e^{x}$ use this fact together with the definition of the natural log $\ln x$ as the inverse of the function of $e^{x}$ to derive an equation for the derivative of $\ln x$

Alexandra Richardson 2022-07-23

Conversion of bases with logarithms
The question says if $\log_{6} (2)$ is $a$ and $\log_{5} (3)$ is $b$ , express $\log_{5} (2)$ in terms of $a$ and $b$
I have tried the change of base formula for $a b$ to no avail, can someone give me a hint to get started, and the solution, hidden behind a spoiler tag, if I get stuck? Thanks.

Luz Stokes 2022-07-23

Solve the equation for exact answer. $9^{3 x + 5} = 18$

yasusar0 2022-07-23

Solve for two variables, two equations with exponents
Solve for both k and x, where $5 = k (300)^{x}$ and $80 = k (600)^{x}$

Greyson Landry 2022-07-23

Can this log question be simplified?
$2^{l o g_{3} 5} - 5^{l o g_{3} 2} .$ I don't know any formula that can apply to it or is there a formula?? Even a hint will be helpful.

Brenton Dixon 2022-07-22

Convert a linear scale to a logarithmic scale
Given a number n, how would I convert this number into a logarithmic scale?
My logarithmic scale would range from 0 to 255 (I'm working with RGB colours), and I would expect values of n from 1 to 1,000,000.
Apologies if this is an easy question.

capellitad9 2022-07-22

If $2 \cdot l o g_{e} (x - 2 y) = l o g_{e} y + l o g_{e} x$ , then find the numerical value of $\frac{x}{y}$
My try:
$2 \cdot l o g_{e} (x - 2 y) = l o g_{e} y + l o g_{e} x$
$l o g_{e} (x - 2 y)^{2} = l o g_{e} x y$
$(x - 2 y)^{2} = x y$
But expanding the LHS doesn't give the result. What should I do further?

Livia Cardenas 2022-07-22

L'Hôpital's as x tends to infinity
I'm searching for the explanation to the limit of:
$lim_{x \to \infty} x \ln \frac{x + 1}{x - 1} .$
I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a section with l'Hôpital.

Parker Bird 2022-07-22

How I could show that : $\log 1 = 0$ ?
I would be like somone to show me or give me a prove for this :
Why $\ln 1 = 0$ ?
Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$ .
Thanks for any replies or any comments!

Marisol Rivers 2022-07-21

Solution for this Logarithmic Equation
Recently I was going through a problem from the book Problems in Mathematics - V Govorov & P Dybov .
$(x - 2)^{\log^{2} (x - 2) + \log (x - 2)^{5} - 12} = 10^{2} \log (x - 2)$
I tried solving by first considering $\log (x - 2)$ as a variable, say $t$ . Then I expressed $(x - 2)$ as $10^{t}$ . Then after using some properties of log, I reached till here-
$10^{t^{3} + 5 t^{2} - 12 t} = 10^{2} t$
or
$10^{t^{3} + 5 t^{2} - 12 t - 2} = t$
Now I have no idea how to approach further. The answer in the references says $x = 3, 102, 2 + 10^{- 7}$

Elsa Brewer 2022-07-21

Proof of logarithmic identity $\log_{g} x = \log_{a} x \cdot \log_{g} a$
I have to prove the alleged link between the logarithms in base g and a
$\log_{g} x = \log_{a} x \cdot \log_{g} a$
I know that this can be written as:
$\frac{\ln x}{\ln g} = \frac{\ln x}{\ln g}$
But does this prove the alleged link or do i have to do something further?
I know that this is derived from:
$\log_{g} x = \frac{\ln x}{\ln g}$

Lorelei Patterson 2022-07-21

How to solve this kind of equation $(x^{y} = y^{x})$ $(x^{y} = y^{x})$
I'm little bit stuck with this system of equations :
$x^{y} = y^{x}$ and $x^{3} = y^{2}$
An obvious solution is $(x, y) = (1, 1)$ but what about the solution $(9 / 4, 27 / 8)$ ?
I know the relation $a^{r} = e^{r \log a}$ but it doesn't help me.
Thanks

detineerlf 2022-07-21

How does exponentiation relate to multiplication?
My book derives the logarithm function as a definite integral of $1 / x$ and defines the exponential function as its inverse. It then extends this definition to other bases:
$b^{x} = e^{\ln (b) x}$
But then somehow we get this really nice property:
$b^{x} = \underset{x times}{\underset{⏟}{b \cdot b \cdot \dots \cdot b}}$
if $x$ is an integer.
This property is too nice to just be a coincidence. Is there a deeper reason this property emerges? I mean it's hard to see how functions derived from relatively complex operations on $1 / x$ (integration) would somehow simplify to such a simple definition for integer arguments.

stylaria3y 2022-07-21

What log rule was used to simply this expression?
I'm unclear how the left side is equal to the right side.
$365 \log (365) - 365 - 305 \log (305) + 305 - 60 \log (365) = 305 \log (\frac{365}{305}) - 60$
I know $\log (a) - \log (b) = \log (a / b)$ but if you stick constants before each ln() then how do you apply the rule to get 305 as the constant on the right side of the equation?

Roselyn Daniel 2022-07-21

Determining $lim_{n \to \infty} {(n^{\frac{1}{n}} - 1)}^{n}$ with only elementary math
I am trying to find this limit:
$lim_{n \to \infty} {(n^{\frac{1}{n}} - 1)}^{n},$
I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you

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