Lesson 9

What is a Logarithm?

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Problem 1

For each equation in the left column, find in the right column an exact or approximate value for the unknown exponent so that the equation is true.

A:

\(10^y = 10\)

B:

\(10^y = 20\)

C:

\(10^y = 2,\!000\)

D:

\(10^y = 900\)

E:

\(10^y = 4\)

1:

0.602

2:

-1

3:

1

4:

2.954

5:

1.301

6:

3.301

7:

1.999

Solution

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Problem 2

Here is a logarithmic expression: \(\log_{10}100\).

  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?

Solution

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Problem 3

The base 10 log table shows that the value of \(\log_{10} 50\) is about 1.69897. Explain or show why it makes sense that the value is between 1 and 2.

Solution

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Problem 4

Here is a table of some logarithm values.

  1. What is the approximate value of \(\log_{10}(400)\)?
  2. What is the value of \(\log_{10}(1000)\)? Is this value approximate or exact? Explain how you know.
\(x\) \(\log_{10} (x)\)
200 2.3010
300 2.4771
400 2.6021
500 2.6990
600 2.7782
700 2.8451
800 2.9031
900 2.9542
1,000 3

Solution

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Problem 5

What is the value of \(\log_{10}(1,\!000,\!000,\!000)\)? Explain how you know.

Solution

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Problem 6

A bank account balance, in dollars, is modeled by the equation \(f(t) = 1,\!000 \boldcdot (1.08)^t\), where \(t\) is time measured in years.

About how many years will it take for the account balance to double? Explain or show how you know.

Solution

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(From Unit 4, Lesson 8.)

Problem 7

The graph shows the number of milligrams of a chemical in the body, \(d\) days after it was first measured.

Coordinate plane, x, time, days, y medicine, milligrams. Curve drawn through 0 comma 20, 1 comma 2 point 5.
  1. Explain what the point \((1,2.5)\) means in this situation.
  2. Mark the point that represents the amount of medicine left in the body after 8 hours.

Solution

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(From Unit 4, Lesson 3.)

Problem 8

The exponential function \(f\) takes the value 10 when \(x = 1\) and \(30\) when \(x = 2\). 

  1. What is the \(y\)-intercept of \(f\)? Explain how you know.
  2. What is an equation defining \(f\)?

Solution

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(From Unit 4, Lesson 6.)