query: If $f(xy) = f(x) + f(y)$, show that $f(.)$ can only be a logarithmic function.
positive_passages.docid: 81799
positive_passages.text: Examples of functions where $f(ab)=f(a)+f(b)$
negative_passages.docid: 14578
negative_passages.text: Proving that an additive function $f$ is continuous if it is continuous at a single point
negative_passages.docid: 145265
negative_passages.text: A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
negative_passages.docid: 41226
negative_passages.text: Find $f:f(xf(x)+f(y))=f(x)^2+y$
negative_passages.docid: 48925
negative_passages.text: BMO1 2009/10 Q5 functional equation: $f(x)f(y) = f(x + y) + xy$
negative_passages.docid: 5373
negative_passages.text: Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.
negative_passages.docid: 15809
negative_passages.text: The problem of x = ln(x)
negative_passages.docid: 103251
negative_passages.text: Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions...)
negative_passages.docid: 30835
negative_passages.text: Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$
negative_passages.docid: 121455
negative_passages.text: $f(x)$ if $f(xy)=f(x) +f(y) +\frac{x+y-1}{xy}$
negative_passages.docid: 124139
negative_passages.text: If $f(f(x))$ is a linear function, must $f(x)$ be linear?
negative_passages.docid: 48592
negative_passages.text: Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
negative_passages.docid: 131586
negative_passages.text: Hard functional equation: $ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $
negative_passages.docid: 26207
negative_passages.text: If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
negative_passages.docid: 128157
negative_passages.text: Textbook clarification: $\log = \ln$
negative_passages.docid: 37966
negative_passages.text: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
negative_passages.docid: 68409
negative_passages.text: Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$