Skip to main content

Lab Manual for MTH 111: Precalculus I: Functions

Mathematics Faculty

Section 2.3 Properties of Logarithms

In this section, we’ll investigate some important properties of logarithms, which will help us to be able to solve exponential and logarithmic equations.

Subsection Textbook Reference

This relates to content in §3.5 of Algebra and Trigonometry 2e
 1 
http://tiny.cc/111Z-Textbook
.

Exercises Preparation Exercises

1.

Simplify each numerical expression without a calculator. Then state the corresponding exponent rule.
(a)
  • \(\displaystyle 8^1\)
  • \(\displaystyle b^1\)
(b)
  • \(\displaystyle 7^0\)
  • \(\displaystyle b^0\)
(c)
  • \(\displaystyle 6^{-2}\)
  • \(\displaystyle b^{-n}\)
(d)
  • \(\displaystyle 5^2 \cdot 5^9\)
  • \(\displaystyle b^n \cdot b^m\)
(e)
  • \(\displaystyle \dfrac{4^7}{4^3}\)
  • \(\displaystyle \dfrac{b^n}{b^m}\)
(f)
  • \(\displaystyle \left(3^2\right)^5\)
  • \(\displaystyle \left(b^n\right)^m\)

2.

Evaluate each logarithm without a calculator.
(a)
\(\log_5(1)\)
(b)
\(\log_2(2)\)
(c)
\(\log(1)\)
(d)
\(\ln(e)\)

Exercises Practice Exercises

1.

Expand \(\log_6(5x^3y)\) as much as posible by rewriting the expression as a sum, difference, or product of logs or constant factors.

2.

Rewrite \(\dfrac{1}{2}\ln(x+5) - 6\ln(x)\) as a single logarithm.

3.

Find the exact value of \(\log_5(75) - \log_5(3) + \log_2(16)\) without using a calculator.

4.

Saskia and José disagree. Saskia says that \(\log_3(x+2) - \log_3(x+1) = \dfrac{\log_3(x+2)}{\log_3(x+1)}\text{,}\) but José says that’s wrong. Who is correct and why?

Subsection Definitions

Definition 2.3.1. Four Properties of Logarithms.

Given any real number \(x\) and any positive number \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{align*} \log_b(1) \amp = 0\\ \log_b(b) \amp = 1\\ \log_b(b^x) \amp = x\\ b^{\log_b(x)} \amp = x \end{align*}

Definition 2.3.2. Product Rule for Logarithms.

Given any positive real numbers \(M\text{,}\) \(N\text{,}\) and \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{equation*} \log_b(M \cdot N) = \log_b(M) + \log_b(N) \end{equation*}

Definition 2.3.3. Quotient Rule for Logarithms.

Given any positive real numbers \(M\text{,}\) \(N\text{,}\) and \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{equation*} \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \end{equation*}

Definition 2.3.4. Power Rule for Logarithms.

Given any real number \(n\text{,}\) positive real numbers \(M\) and \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{equation*} \log_b(M^n) = n \cdot \log_b(M) \end{equation*}

Definition 2.3.5. Change of Base Formula.

Given positive real numbers \(M\text{,}\) \(n\text{,}\) and \(b\text{,}\) with \(b \neq 1\) and \(n\neq 1\text{,}\)
\begin{equation*} \log_b(M) = \frac{\log_n(M)}{\log_n(b)} \end{equation*}

Exercises Exit Exercises

1.

(a)
Only one of these is true. Which one and why?
  1. \(\displaystyle \log_2(8) + \log_2(16) = \log_2(8 \cdot 16)\)
  2. \(\displaystyle \log_2(8) \cdot \log_2(16) = \log_2(8 + 16)\)
(b)
Only one of these is true. Which one and why?
  1. \(\displaystyle \dfrac{\log_3(27)}{\log_3(9)} = \log_3(27-9)\)
  2. \(\displaystyle \log_3(27) - \log_3(9) = \log_2(\frac{27}{9})\)
(c)
Rewrite each of these as a single logarithm.
  1. \(\displaystyle \log(10) - \log(2) + \log(3)\)
  2. \(\displaystyle \ln(a) - \ln(b) - \ln(c)\)
(d)
Fully expand each of the following.
  1. \(\displaystyle \log_2\left(\dfrac{x^2}{y^3z}\right)\)
  2. \(\displaystyle \ln(e^5n^3\sqrt[4]{k})\)

Reflection Reflection

1.

On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
PrevTopNext