All Questions
Tagged with algebra-precalculus logarithms
1,541
questions
-2
votes
1
answer
62
views
What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]
I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
3
votes
2
answers
279
views
A problem that could use substitution or logs, not sure which works better
This is one of those brain teaser problems on instagram, and it starts here:
$$x^{x^2-2x+1} = 2x + 1$$
And we want to solve for x. My first instinct was to try this
$$\ln(x^{x^2-2x+1}) = \ln(2x + 1)\\
...
- 2,692
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\sqrt{x+1}=-2$$
And obviously there is no value for it,
However,
If you do the thing with $e$ and $\ln{}$
$$e^{\ln{\sqrt{x+1}}}$$
and
$$e^{\frac{1}{2}\cdot (\ln{x+1})}$$
Then ...
- 19
0
votes
1
answer
41
views
How to solve for a value in a log
I have a formula:
Weight=onerepmax*(0.488 + 0.538 * ln(-0.075*reps))
And I need to solve for reps given a onerepmax and a weight.
I got as far as:
...
- 113
-5
votes
2
answers
85
views
If the domain of $f(x)$ is $(-3, 1)$, then what is the domain of $f(\ln x)$? [closed]
I need a clear explanation for this question:
If the domain of $f(x)$ is $(-3, 1)$ then the domain of $f(\ln x)$ is ...
a) $\;(e^{-1}, e^3)$
b) $\;(0, \infty)$
c) $\;(1, \infty)$
d) $\;(e^{-3}, e^...
0
votes
6
answers
195
views
How would you prove $\log_{2}x < \sqrt x$ for $x > 16$? [closed]
I'm not really showing how to prove this, since I tried finding the $x$-intercepts/zeros of $f(x) = \sqrt x - \log_{2} x$ , and see that $x = 4, 16$ work but inspection, but I'm not sure how to ensure ...
- 883
-3
votes
2
answers
191
views
How do you solve this equation $ \log_{2}(x) = \sqrt x$? [closed]
Disclaimer: Guys before voting to get the question closed I strongly feel we should instead have a feature on MSE that can merge such similar/duplicate questions since we got some really cool/through ...
- 883
2
votes
1
answer
90
views
Solving $\frac{\ln(y/x)}{y-x} = t$ for $x$ [duplicate]
I am having trouble solving an algebra formula which is for a project of mine.
I must solve for $x$ ($y$ is a known value).
$$\frac{\ln\left(\dfrac{y}{x}\right)}{y-x} = t$$
As I try to solve the ...
- 29
1
vote
1
answer
41
views
What to consider when taking kth root on both sides of equality
Say I have the following expression:
$10^{l} = a^{k}$
If I take the kth root of both sides, does that mean we get:
$10^{\frac{l}{k}} = a$
We don't have to consider anything with plus or minus?
- 883
1
vote
2
answers
72
views
Log X to what base n yields a whole number [closed]
Does there always exist a real number 'n' such that $log_{n}x$ is a whole number for any real number x?
If yes what would the function to find this number look like?
- 11
0
votes
0
answers
34
views
Why is there no logarithmic form of the exponential distributive rule/power of a product rule?
When learning the laws of exponents and logarithms, one finds that there is a correspondence. Each law of exponents has a corresponding equivalent expression in terms of logarithms. For example, the ...
- 16.9k
1
vote
4
answers
922
views
Why roots aren't the inverse of exponentiation but logarithms?
I think it's easy to see it when we look at the inverse of the function "$f(x) = a^x$" but I wonder if there's other way to look at it besides just analyzing the function. I was taught my ...
- 21
1
vote
5
answers
101
views
If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?
So, I've tried using the properties:
$$\log_a b = \frac{\log_c b}{\log_c a}$$
and..
$$\log_a bc = \log_a b + \log_a c$$
And, the final simplification should be in the following options:
$$A. \frac{1-a+...
- 13
3
votes
2
answers
153
views
Logarithmic inequality involving $a_1, a_2, ..., a_n$
Given the real numbers $a_1, a_2,...,a_n$ all greater than $1$, such that $\prod_{i=1}^{n} a_i=10^n$, prove that:
$$\frac{\log_{10}a_1}{(1+\log_{10}a_1)^2}+\frac{\log_{10}a_2}{(1+\log_{10}a_1 + \log_{...
- 409
4
votes
1
answer
132
views
Reducing product of powers of logarithm
I am trying to show that
$$(\log(a))^n (\log(b))^m = P(\log(a^ib^j)), \quad i,j \in \{-1,0,1\}$$
where $P$ is a polynomial and $n \ge m \ge 1$ are natural numbers. Using Binomial identities for the ...
- 3,360
1
vote
3
answers
115
views
How to solve $x+1=5e^{4x}$ [closed]
How to solve $x+1=5e^{4x}$
In general, I know to take ln() of both sides to bring down the exponent for e, but the left side is also a variable.
- 4,771
0
votes
1
answer
53
views
Solving a logarithmic equation with different logarithmic exponents.
I had a logarithmic equation which originally was
original
https://i.sstatic.net/2fSf1jNM.png
$$5^{\log_{10}x}-3^{\log_{10}x-1}=3^{\log_{10}x+1}-5^{\log_{10}x-1}$$
but I thought that this should also ...
3
votes
2
answers
77
views
How to evaluate an expression of higher powers and roots using logarithms?
I am struggling with the following question from a Dutch algebra exam from the 1950s. The instructions are as follows:
Calculate with logarithms.
$$
x = \frac{\sqrt[3]{(23.57^2 - 15.63^2)}}{{0....
-2
votes
1
answer
59
views
How does $\log(y)=C+t$ become $y = C e^{t}$? [closed]
I came across this transformation :
$$\begin{align}
\log(y) &= C + t \tag{1} \\[4pt]
y &= C e^{t} \tag{2}
\end{align}$$
How was the first step simplified into the second?
- 7
0
votes
2
answers
69
views
Suppose a colony of cells starts with 10 cells, and their number triples every hour. After how many hours will there be 500 cells?
I thought it would be log(500), which gives approximately 2.69897. I know that there could be alternative forms of the answer, but for the life of me, I don't understand how they arrive at this ...
- 83
0
votes
1
answer
20
views
Question regarding finding the rate of depreciation in a half-life equation. This uses logarithms but I am unsure how to arrive at the answer given. [closed]
This question is from a chapter regarding logarithms. I have correctly answered part (a) of the question, which simply involves substituting the variables for the values given, so I need no help with ...
- 83
-1
votes
1
answer
27
views
A star's magnitude $M$ and intensity $I$ satisfy $M=6-2.5\log\frac{I}{I_0}$. Find the ratio of intensities between stars of magnitude $1$ and $3$. [closed]
My textbook doesn't go into how to solve this question.
The magnitude of a star is given by the equation
$$M=6-2.5\log\frac{I}{I_0}$$
where $I_0$ is the measure of the faintest star, and $I$ is the ...
- 83
10
votes
2
answers
797
views
Question regarding nature of logarithmic equations
While reading my textbook's chapter about logarithms and seeing the solved examples I noticed in various places that the author was able to make the $\log$ just disappear in a equation or inequality ...
- 356
1
vote
2
answers
30
views
Getting the domain of a real function with iterated logarithms [duplicate]
I would like to find the domain of the function
$$f(x)\:=\: \log_4\,\log_5\,\log_3\big(\,18x - x^2 - 77\,\big)$$
as a subset of $\mathbb R\,$.
I looked at the solution of the above problem, and it ...
- 356
0
votes
1
answer
68
views
Wrong simplification of $2^{\sqrt{\log_2n}}$ [duplicate]
I am trying to do the exercise 01, chapter 02 of the book: Algorithm Design [Kleinberg _ Tardos] - publication version 03 of the book
I need to manipulate $2^{\sqrt{\log_2n}}$, what I did was:
$2^{\...
1
vote
3
answers
131
views
Solve $x^2-2x+1=\log_2( \frac{x+1}{x^2+1})$
Solve in $\mathbb R$ the following equation $$x^2-2x+1=\log_2 (\frac{x+1}{x^2+1})$$
First of all from the existence conditions of the logarithm, we have $x > -1$. Analyzing $x^2 - 2x - 1$ , we get ...
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0
votes
1
answer
76
views
How Can Solving $\left( 2n \right)^{\left( \log_b 2 \right)} = \left( 5n \right)^{\left( \log_b 5 \right)}$ be Generalized for any Base of $\log_b n$?
The title of the question Solving $(2n)^{\log 2}=(5n)^{\log 5}$ asks how to solve (in some western conventions) base 10 for $(2n)^{\log 2}=(5n)^{\log 5}$. However that question's title does not seem ...
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votes
0
answers
343
views
An analytic solution to solve $x^9=3^x$
I want to find a way to solve $x^9=3^x$ analytically, for two roots. one of them can be found below $$x^9=3^x\\(x^9)^{\dfrac {1}{9x}}=(3^x)^{\dfrac {1}{9x}}\\x^ { \ \frac 1x}=3^{ \ \frac 19}\\x^ { \ \...
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1
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1
answer
90
views
Simplify $\log((10\cdot 8 )^{\frac{1}{2}} \times (0.24)^{\frac{5}{3}} \div (90)^{-2})$
Simplify $\log((10\cdot 8 )^{\frac{1}{2}} \times (0.24)^{\frac{5}{3}} \div (90)^{-2})$
$\Rightarrow \log(10\cdot 8)^{\frac{1}{2}}+\log(0.24)^{\frac{5}{3}}-\log(90)^{-2} \tag{1}$
$\Rightarrow \dfrac{1}...
1
vote
1
answer
68
views
Find the value of $\sqrt[5]{0.00000165}$ given $\log165=2.2174839$ and $\log697424=5.8434968$
Find the value of $\sqrt[5]{0.00000165}$ given $\log165=2.2174839$ and $\log697424=5.8434968$
$\log x=\log\sqrt[5]{0.00000165}$
$\Rightarrow \log x =\dfrac{1}{5}\log0.00000165=\dfrac{1}{5}(\overline{...
3
votes
6
answers
374
views
Solving $(2n)^{\log 2}=(5n)^{\log 5}$
I have seen this equation from a link named Asisten and German Academy, (it is a video of Facebook) where there is a complicate solution (I invite to watch it) for
$$(2n)^{\log 2}=(5n)^{\log 5}$$
I ...
- 7,792
0
votes
1
answer
67
views
Solutions to Some Logarithmic Inequalities
Suppose we have an inequation as shown below:$$I_0:\space \ln (x) > \frac{x-2}{x}$$ Now we would like to find the largest set $S$ of real numbers such that any element $p\in S$ will satisfy $I_0$ ...
3
votes
0
answers
78
views
What functions satisfy $f(ax) - f(a(x-1)) > f(b(x+1)) - f(bx)$ for all $a, b \in \mathbb{R}^+$ and $x \in \mathbb{Z}^+$.?
I am looking at a family of functions $f : [0, \infty) \rightarrow [-\infty, \infty)$ satisfying the following property:
$$f(bx) - f(b(x-1)) > f(a(x+1)) - f(ax) \quad \text{for all $a, b \in \...
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4
votes
2
answers
157
views
Why does $x^{\log_a y} = y^{\log_a x}$ ? (intuitive reason)
So I have seen mathematical proofs for this. They used other logarithmic identities to get this result. I am looking for a more intuitive approach to this and not just equations which lead to this.
...
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1
vote
0
answers
111
views
Help Solving a logarithmic equation $P\times\log{(1-\frac{a}{nP})} = -b\times\log{(1+\frac{c}{n})}$ for P where P>0
I have tried using algebraic Logarithm and exponent rules but I cannot get P into a common form. I get P in exponent and standard form or I get P in Logarithmic and standard form
My attempt so far:
$...
- 11
1
vote
1
answer
111
views
How to prove $x^{\ln x} > \frac{x}{2} + \frac{1}{2x} $?
How to prove the following? $$x^{\ln x} > \dfrac{x}{2} + \dfrac{1}{2x} \tag{1} $$ for all $x \in \mathbb{R}^+\setminus \{1\}$?
I could prove $ x^{\ln x} > x $ and
$ x^{\ln x} > x/2 $ (in ...
- 3,227
1
vote
2
answers
141
views
Simplifying logarithmic expression
Background
An integral is solved, and I get the following expression:
$$I = \frac12 \ln \left| \frac{u-1}{u+1} \right| - \frac u{u^2 - 1} + C$$
Teacher's solution states that from there:
If we set $u ...
- 4,124
2
votes
5
answers
154
views
Given that $A=\text{log}_{16}15$ and $B=\text{log}_{12}18$, find $\text{log}_{25}24$ in terms of A and B.
Given that $A=\text{log}_{16}15$ and $B=\text{log}_{12}18$, find $\text{log}_{25}24$ in terms of $A$ and $B$.
I found that the answer is $\frac{B-5}{-2AB-2A+4B-2}$ but I used a very inefficient steps ...
- 285
0
votes
0
answers
50
views
Solution of Two Functions
Given $f(x) = -1 + 5(1.02)^x$ and $g(x) = \ln(3 - x)$, for what value of $x$ does $f(x) = g(x)$? I have been trying to solve this question for quite some time and I always seem to hit a dead end. What ...
- 11
2
votes
1
answer
128
views
Find the value of $\log_{5}(0.0016)$
Find the value of $\log_{5}(0.0016)$
$y=\log_{5}(0.0016)=\log_{5}(0.2)^4 \Rightarrow 5^y=(0.2)^4$
$\Rightarrow \log5^y=\log(0.2)^4$
$\Rightarrow y\log5=4\log(\dfrac{1}{5})$
$\Rightarrow y\log5=4(\...
0
votes
0
answers
30
views
Show that for $a \neq b$ it holds: $\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$ [duplicate]
Show that for $a \neq b$ it holds:
$$\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$$
My first idea was to rearrange
$$2 \cdot (e^b-e^a) < (b-a)(e^b+e^a)$$
$$2e^b-2e^a < be^b + be^a - ae^b -e^a$$
...
0
votes
1
answer
124
views
Solving $\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$ [closed]
Here's the question I came across, they're inverses in this case, but I imagine that there is a way to do that without them being inverses.
$$\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$$
1
vote
0
answers
108
views
Solving $\frac{8^x-2^x}{6^x-3^x} = 2$ [duplicate]
$\frac{8^x-2^x}{6^x-3^x} = 2$
I was able to use difference of cubes on the numerator, and setting $a= 2^x$ would give the following:
$\frac{a(a-1)(a+1)}{3^x(a-1)} = 2$
Now we have a common factor and ...
- 1,190
4
votes
1
answer
262
views
Is $e^{\ln(-7) }= -7$?
I was going over some homework, and stumbled across a true or false question that presented as the following:
$e^{\ln(-7)}$ = -7, True or False
Seeing as the definition of a logarithm presents the ...
- 43
1
vote
1
answer
71
views
Solving Logarithmic Expression
In the context of the thermodynamics of mixing two separate gases at the same temperature and pressure, one has the generic equation,
\begin{gather*}
-\frac{\Delta S_{mix}}{nR} = X_A \ln(X_A) + X_B \...
- 165
0
votes
0
answers
43
views
Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
- 185
1
vote
1
answer
48
views
Showing $ (2/3)^{\log_{3/2}(n/n_0)} =n_0/n $
I read the following identity in the CLRS Introduction to Algorithms book, and I can't work out the computation.
$$
(2/3)^{\log_{3/2}(n/n_0)} =n_0/n
$$
I tried to expand the exponent using the ...
- 185
1
vote
0
answers
88
views
Does $\log\left(\frac{A}{B}\right)$ really equal to $\log(A)-\log(B)$? [duplicate]
I was investigating the laws of logarithm and playing with Desmos when I realized something curious.
The example equation is $f(x)=\log\left(\frac{2x-4}{6x-8}\right)$ and the graph is this:
The law ...
- 11
1
vote
1
answer
65
views
Does this equation have a closed-form solution for $f(x) = 0$?
I have the equation $f(x) = -n^{-x} × (n + 1)^{x - n - 2} × \left(\left((n + 1)^{n + 1} - n^n × x\right) × (\ln(n) - \ln(n + 1)) + n^n\right)$ where $n$ is a positive real number and $\ln(z)$ is the ...
- 1,861
1
vote
3
answers
263
views
Can the "simple" equation $e^x=\log(x)$ be solved using algebra?
I came across this really simple-looking yet astonishingly hard problem to solve:
$$e^x=\log(x).$$
I tried to use Lambert-W function, but I cannot get it to the required standard form. Even Wolfram ...
- 13