Logarithms? Trigonometric functions? Dostoevski? Has Bill Bennett lost his mind? Is he seriously proposing that American high school students be required -- not just encouraged but required -- to study this stuff?

The secretary of education is dead serious, and he has not lost his mind at all. He is convinced that every high school student -- not just the gifted, college-bound students, but every student from the ninth through the 12th grade -- should be exposed to a tough and demanding core curriculum. His recommendations make great good sense.

Nearly five years have passed since the National Commission on Excellence in Education brought forth its stunning report, ''A Nation at Risk.'' That report documented and denounced the ''rising tide of mediocrity'' that had overwhelmed public education. It fired up the boilers in many state legislatures, and here and there it produced real improvements.

Then the steam leaked out. Most of the states have significantly improved teacher salaries; a few have opted for systems of merit pay. But only 13 percent of the school districts have instituted minimum competency tests for teachers. None has extended the school year, and only three states -- Florida, Louisiana and Pennsylvania -- have mandated the kind of required courses that Bennett recommends.

Public education in America is in serious trouble. The taxpayers are supporting 17,000 high schools with an enrollment of 12 million teen-agers, and with some notable exceptions these young people are being shortchanged. They are not getting the education that will equip them for the competitive world of the 21st century.

Maybe Bennett's statement will inspire state legislators to mandate reforms. In Bennett's ideal ''James Madison High School,'' students would be required to take four years of English, three years each of social studies, math and science, two years each of a foreign language and physical education, and a half-year each of art and music. At every step along the way, students would be required to write papers. They would read serious books and plays. They would get a solid grounding in history and government. They would be expected to master algebra and trigonometry, and they would have an opportunity to study calculus.

The schedule would leave abundant room, especially in the senior year, for electives -- for band, typing, computer operations, driver education and the like. Bennett emphatically challenges the notion that he is promoting elitism. He finds the charge insulting to poor blacks and Hispanics who may be culturally disadvantaged. Of course they can learn, he says, and he cites examples to prove it.

At New York City's A. Philip Randolph High School, half of the 1,500 black and Hispanic students come from poor families, but they are thriving on a core curriculum even tougher than Bennett's ideal. In Los Angeles, the James A. Garfield High School serves 3,500 students in the East Los Angeles barrio, 90 percent of whom are disadvantaged. Seventy percent of Garfield's students go on to higher education.

Bennett makes the same point that the Commission on Excellence in Education made in 1983. Our planet shrinks. Our teen-agers will be competing as adults with their opposite numbers around the world. Does he ask too much in the area of mathematics? In an international competition in 1982, our best high school mathematicians finished dead last. Is it unreasonable to require two years of a foreign language? In France, students must take six years of one foreign language and four years of another.

We want our students to take from high school, Bennett says, ''a shared body of knowledge and skills, a common language of ideas, a common moral and intellectual discipline.'' Our graduates should know math and science, history and literature. ''We want them to think for themselves, to respond to important questions, to solve problems, to pursue an argument, to defend a point of view, to understand its opposite, and to weigh alternatives.''

This is what education is all about -- the development of those ''habits of mind and traits of character'' that our society prizes. To the extent that our high schools are failing to inculcate those values, they are failing the children, and failing the country as well.