SUPPLEMENTARY EXERCISES WITH LOGARITHMS

By Alex Pintilie

1) Which numbers x satisfy the equation: (log₃x)(log_x5) = log₃5 ?

2) Suppose that the Canadian dollar loses 5% of its value each year. How many years are needed in order that the Canadian dollar to lose 90% of its value ? (That is, the future value of the dollar to become the present value of a dime.)

3) Simplify the product: P = (log₂3)(log₃4)(log₄5) ... (log₃₁32)

4) If p = find a^p

5) If log _b (xy) = 11 and log _b (x/y) = 5, what is log _b x?

6) Positive integers A, B, and C, with no common factor greater than 1, exist such that

A log ₂₀₀ 5 + B log ₂₀₀ 2 = C. What is A + B + C?

7) What is the value of ?

8) A computer manufacturer finds that when x millions of dollars are spent on research, the profit, P(x), in millions of dollars, is given by . How much should be spent on research to make a profit of 40 million dollars?

9) Solve the system of equations and .

10) Solve the equation log₃ (x-2) + log₃ 10 = log₃ (x² + 3x – 10)

11) log₂ (9-2^x) = 3 – x

Answers:

1) all x > 0, x ¹ 1 2) about 45 years 3) P = 5 4) a^p = 2 5) 8 6) 6 7) 5/2 8) 78 million 9) (1/4, - 1) 10) x = 5 11) 0, 3