1950 AMC 12 Problem 18

Question

Of the following

(1)\quad a(x-y)=ax-ay

(2)\quad a^{x-y}=a^x-a^y

(3)\quad \log(x-y)=\log x-\log y

(4)\quad \dfrac{\log x}{\log y}=\log x-\log y

(5)\quad a(xy)=ax\times ay

Only

1

and

4

are true

Only

1

and

5

are true

Only

1

and

3

are true

Only

1

and

2

are true

Only

1

is true

Compare each statement with the standard rules for exponents and logarithms

In particular, subtraction inside an exponent or logarithm does not distribute as written

Statement

1

is the distributive property. Statement

2

is false because

a^{x-y}=\frac{a^x}{a^y}.

Statement

3

is false because

\log x-\log y=\log(\frac{x}{y}).

Statement

4

confuses the change-of-base quotient with a difference, and statement

5

has right side

a^2xy,

not

axy.

Thus, only statement

1

is true, so the correct answer is E.

Problem 18 in other years