Lesson 10

The expression \(\log_{10} 1,\!000 = 3\) can be read as: “The log, base 10, of 1,000 is 3.”

It can be interpreted as: “The exponent to which we raise a base 10 to get 1,000 is 3.”

Take turns with a partner reading each equation out loud. Then, interpret what they mean.

• \(\log_{10} 100,\!000,\!000 = 8\)
• \(\log_{10} 1 = 0\)
• \(\log_2 16 = 4\)
• \(\log_5 25 = 2\)

1. Use the table to find the exact or approximate value of each expression. Then, explain to a partner what each expression and its approximated value means.
   1. \(\log_2 2\)
   2. \(\log_2 32\)
   3. \(\log_2 15\)
   4. \(\log_2 40\)
2. Solve each equation. Write the solution as a logarithmic expression.
   1. \(2^y = 5\)
   2. \(2^y=70\)
   3. \(2^y=999\)

These equations express the same relationship between 2, 16, and 4:

1. Each row shows two equations that express the same relationship. Complete the table.

   	exponential form	logarithmic form
   a.	\(2^1= 2\)
   b.	\(10^0=1\)
   c.		\(\log_3 81 = 4\)
   d.		\(\log_5 1 =0\)
   e.	\(10^\text{-1}=\frac{1}{10}\)
   f.	\(9^\frac12=3\)
   g.		\(\log_2 \frac18= \text - 3\)
   h.	\(2^y=15\)
   i.		\(\log_5 40 = y\)
   j.	\(b^y = x\)

2. Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
   1. “To what exponent do we raise the number 4 to get 64?”
   2. “What is the log, base 2, of 128?”

Many relationships that can be expressed with an exponent can also be expressed with a logarithm. Let’s look at this equation: \(\displaystyle 2^7 = 128\) The base is 2 and the exponent is 7, so it can be expressed as a logarithm with base 2:

\(\log_2 128 = 7\)

In general, an exponential equation and a logarithmic equation are related as shown here:

Exponents can be negative, so a logarithm can have negative values. For example \(3^{\text-4} = \frac{1}{81}\), which means that \(\log_3 \frac{1}{81} = \text-4\).

An exponential equation cannot always be solved by observation. For example, \(2^x = 19\) does not have an obvious solution. The logarithm gives us a way to represent the solution to this equation: \(x=\log_2 19\). The expression \(\log_2 19\) is approximately 4.25, but \(\log_2 19\) is an exact solution.