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Express the given expression as a single logarithm
Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
Apr 10, 2026
Archive: News
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Empty Set $\{\}$ is the Only Basis of the Zero Vector Space $\{0\}$
Question Suppose we want to find a basis for the vector space $\{0\}$. I know that the answer is that the only basis is the empty set. Is this answer a de...
Apr 10, 2026
![Prove $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$](https://news.conceivablytech.com/uploads/prove-x-4-2x-2-4-is-irreducible-in-mathbb-q-x_720.jpg)
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Prove $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$
For some problem from my Galois Theory course, I need to prove that the polynomial $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$. I know it has no roots ...
Apr 10, 2026
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Element-wise multiplication or Hadamard product
$a_{ij}$ and $b_{ij}$ elements of matrix A and B (same dimensions). I want to multiply the matrices element-wise so the resulting matrix $s$ have the same...
Apr 10, 2026
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The equation of the normal when the gradient of the tangent = 0
Q. Find the equation of the tangent and normal to $x^2-xy+y^2 = 3$ at $(1,2)$. I have done the first part. I found: $$\displaystyle\frac{dy}{dx} = \displa...
Apr 10, 2026
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Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$
Prove by induction the summation of $\frac1{2^n}$ is greater than or equal to $1+\frac{n}2$. We start with $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\...
Apr 10, 2026
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Geometrically, what is the span of vectors?
Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions i...
Apr 10, 2026
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"How many public playgrounds exist in the United States?" How to answer using statistics and probability
I have a goal of estimating how many public playgrounds exist in the United States. There are many methods of gathering real data about playgrounds, but, ...
Apr 10, 2026
![Prove that 1+1=2 [duplicate]](https://news.conceivablytech.com/uploads/prove-that-1-1-2-duplicate_720.jpg)
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Prove that 1+1=2 [duplicate]
Possible Duplicate: How do I convince someone that $1+1=2$ may not necessarily be true? I once read that some mathematicians provided a very length proof ...
Apr 10, 2026
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Confusion about Taylor approximation away from center point
I'm trying to learn Taylor expansions and was watching a tutorial here. In the tutorial, Taylor approximation is introduced by first showing Maclauri...
Apr 10, 2026
