Lesson 10

Interpreting and Writing Logarithmic Equations

  • Let’s look at logarithms with different bases.

Problem 1

  1. Use the base-2 log table (printed in the lesson) to approximate the value of each exponential expression.
    1. \(2^5\)
    2. \(2^{3.7}\)
    3. \(2^{4.25}\)
  2. Use the base-2 log table to find or approximate the value of each logarithm.
    1. \(\log_2 4\)
    2. \(\log_2 17\)
    3. \(\log_2 35\)

Problem 2

Here is a logarithmic expression: \(\log_2 64\).

  1. How do we say the expression in words?
  2. Explain in your own words what the expression means.
  3. What is the value of this expression?

Problem 3

  1. What is \(\log_{10}(100)\)? What about \(\log_{100}(10)\)?
  2. What is \(\log_{2}(4)\)? What about \(\log_{4}(2)\)?
  3. Express \(b\) as a power of \(a\) if \(a^2  = b\).

Problem 4

In order for an investment, which is increasing in value exponentially, to increase by a factor of 5 in 20 years, about what percent does it need to grow each year? Explain how you know.

(From Unit 4, Lesson 4.)

Problem 5

Here is the graph of the amount of a chemical remaining after it was first measured. The chemical decays exponentially.

Coordinate plane, horizontal, w, weeks since measurement, vertical, amount of chemical, milligrams. A curve begins at 0 comma 800 and decreases as w increases toward approximately 3 comma 220.

What is the approximate half-life of the chemical? Explain how you know.

(From Unit 4, Lesson 7.)

Problem 6

Find each missing exponent.

  1. \(10^?=100\)
  2. \(10^? = 0.01\)
  3. \(\left(\frac {1}{10}\right)^? = \frac{1}{1,000}\)
  4. \(2^? = \frac12\)
  5. \(\left(\frac12\right)^? = 2\)
(From Unit 4, Lesson 8.)

Problem 7

Explain why \(\log_{10}1 = 0\).

(From Unit 4, Lesson 9.)

Problem 8

How are the two equations \(10^2 = 100\) and \(\log_{10}(100) = 2\) related? 

(From Unit 4, Lesson 9.)