All Questions
102
questions
-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
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
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}$$
How was the first step simplified into the second?
- 7
0
votes
2
answers
69
views
Suppose a colony of cells starts with 10 cells, and their number triples every hour. After how many hours will there be 500 cells?
I thought it would be log(500), which gives approximately 2.69897. I know that there could be alternative forms of the answer, but for the life of me, I don't understand how they arrive at this ...
- 83
1
vote
3
answers
131
views
Solve $x^2-2x+1=\log_2( \frac{x+1}{x^2+1})$
Solve in $\mathbb R$ the following equation $$x^2-2x+1=\log_2 (\frac{x+1}{x^2+1})$$
First of all from the existence conditions of the logarithm, we have $x > -1$. Analyzing $x^2 - 2x - 1$ , we get ...
- 125
1
vote
0
answers
111
views
Help Solving a logarithmic equation $P\times\log{(1-\frac{a}{nP})} = -b\times\log{(1+\frac{c}{n})}$ for P where P>0
I have tried using algebraic Logarithm and exponent rules but I cannot get P into a common form. I get P in exponent and standard form or I get P in Logarithmic and standard form
My attempt so far:
$...
- 11
1
vote
3
answers
325
views
How do I solve $x^{4^x}=4$?
My friend showed me this problem from Twitter and I am struggling to solve it. I see that I can manipulate it into several equations (some of which I'll insert below), but none seem to be any progress ...
- 1,688
1
vote
1
answer
87
views
Solve for $x$: $3^{2x}=5^{x-1}$
So I recently got myself an AP Precalculus book (for anyone who wants to know which book, I will have that in the "To Clarify" section at the bottom of this post) and was looking through ...
- 2,652
1
vote
3
answers
53
views
Substitution of $x=\ln y$
I must be forgetting a log rule, but I'm reading a solution where $$\frac{x^p}{a^x}$$ equals $$\frac{(\ln y)^p}{y^{\ln a}}$$ after substituting $x = \ln y$ but I can't figure out how that works. I can ...
- 73
0
votes
1
answer
99
views
Solving for the number of digits in large exponent problems
I'm studying for state Mathcounts and found this problem solution in the 2014 state test:
How many digits are in the integer
representation of 2^30?
Looking at the powers of 2, we have
2, 4, 8, 16, ...
7
votes
2
answers
182
views
Solving for x in logarithmic equation $\log_4(2x) = \frac{1}{2}x^2 - 1$
I am trying to solve for $x$ in the equation $\log_4(2x) = \frac{1}{2}x^2 - 1$. I have tried converting the logarithmic expression to exponential form, but I am not able to isolate $x$ in the ...
0
votes
1
answer
2k
views
Is there a "Exponential Form" of the "Logarithmic Change of Base"?
There might be no "answer" but I figured it was worth asking the community.
NOTE I wanted to share what ChatGPT thought about this, ChatGPT responded with:
The logarithmic formula $\log_{b}...
- 183
-1
votes
1
answer
79
views
All real and closed-form roots of $\log_2x=\frac {2^{x-1}}{x}$
What are the closed-form roots of
$$\log_2x=\frac {2^{x-1}}{x}$$
?
My attempts:
Closed-form means, I assume that the Lambert W function can work.
I know that, at least $x>0$.
Wolfram Alpha gives ...
- 1,659