# Topic 17 Logarithmic Functions

## 17.1 Estimate the Number of Digits

Can you estimate the number of digits in the integer part of the number \(2^{15}\times \sqrt{2020}\div 2021\)?

## 17.2 Definition and Graphs of Logarithmic Function

For \(x>0\), \(b>0\) and \(b\neq 1\), there is a unique number \(y\) satisfying the equation \(b^y=x\). We denote the unique number \(y\) by \(\log_bx\), read as logarithm to the base \(b\) of \(x\). In other words, the defining relation between exponentiation and logarithm is \[ y=\log_bx \quad\text{if and only if} \quad b^y=x. \] The function \(f(x)=\log_bx\) is called the logarithmic function \(f\) of \(x\) with the base \(b\).

Graphs of logarithmic functions:

## 17.3 Common Logarithms and Natural Logarithms

A logarithmic function \(f(x)\) with base 10 is called the common logarithmic function and denoted by \(f(x)=\log x\).

A logarithmic function \(f(x)\) with base the natural number \(e\) is called the natural logarithmic function and denoted by \(f(x)=\ln x\).

## 17.4 Basic Properties of Logarithms

When \(b>0\) and \(b\neq 1\), and \(x>0\), we have

- \(b^{\log_bx}=x\).
- \(\log_b(b^x)=x\).
- \(\log_bb=1\) and \(\log_b1=0\).

**Example 17.1 **
Convert between exponential and logarithmic forms.

- \(\log x=\frac{1}{2}\)
- \(3^{2x-1}=5\)

*Solution. *

When converting between exponential and logarithmic forms, we move the base from one side to the other side, then add or drop the log sign.

- Move the base 10 to the right side and drop the log from the left: \[x=10^{\frac{1}{2}}\]
- Move the 3 to the right and add log the the right: \[2x-1=\log_35\]

**Example 17.2 **
Evaluate the logarithms.

- \(\log_42\)
- \(10^{\log(\frac{1}{2})}\)
- \(\log_5(e^0)\)

*Solution. *

The key is to rewrite the log and the power so that they have the same base.

- \(\log_42= \log_44^{\frac{1}{2}}=\frac{1}{2}\).
- \(10^{\log\frac{1}{2}}=10^{\log_{10}\frac{1}{2}}=\frac12\)
- \(\log_5(e^0)=\log_51=0\)

**Example 17.3 **
Find the domain of the function \(f(x)=\ln(2-3x)\).

*Solution. *

The function has a real output if \(2-3x>0\). Solving the inequality, we get \(x<\frac{2}{3}\). So the domain of the function is \((-\infty, \frac{2}{3})\).

## 17.5 Properties of Logarithms

For \(M>0\), \(N>0\), \(b>0\) and \(b\neq 1\), we have

- (The product rule) \(\log_b(MN)=\log_bM+\log_bN\)
- (The quotient rule) \(\log_b(\frac MN)=\log_bM-\log_bN\).
- (The power rule) \(\log_b(M^p)=p\log_bM\), where \(p\) is any real number.
- (The change-of-base property) \(\log_bM=\dfrac{\log_aM}{\log_ab}\), where \(a>0\) and \(a\neq 1\). In particular, \[ \log_bM=\dfrac{\log M}{\log b} \quad\text{and}\quad \log_bM=\dfrac{\ln M}{\ln b}. \]

**Example 17.4 **
Expand and simplify the logarithm \(\log_2\left(\frac{8\sqrt{y}}{x^3}\right)\).

*Solution. *

\[ \begin{aligned} \log_2\left(\frac{8\sqrt{y}}{x^3}\right)&=\log_2(8\sqrt{y})-\log_2(x^3)\\ &=\log_28+\log_2(y^{\frac{1}{2}})-3\log_2x\\ &=3+\frac{1}{2}\log_2y-3\log_2x. \end{aligned} \]

**Example 17.5 **
Write the expression \(2\ln(x-1)-\ln(x^2+1)\) as a single logarithm.

*Solution. *

\[ 2\ln(x-1)-\ln(x^2+1)=\ln((x-1)^2)-\ln(x^2+1)=\ln\left(\frac{(x-1)^2}{x^2+1}\right). \]

**Example 17.6 **
Evaluate the logarithm \(\log_34\) and round it to the nearest tenth.

*Solution. *

On most scientific calculator, there are only the common logarithmic function `LOG`

and the natural logarithmic function `LN`

. To evaluate a logarithm based on a general number, we use the change-of-base property. In this case, the value of \(\log_34\) is \[\log_34=\frac{\log4}{\log3}\approx 1.3.\]

**Example 17.7 **
Simplify the logarithmic expression
\[
\log_2(x^{\log 3})\log_32.
\]

*Solution. *

\[ \log_2(x^{\ln 3})\log_32 =(\ln 3\log_2x)\log_32=\ln3\left(\frac{\ln x}{\ln 2}\right)\left(\frac{\ln 2}{\ln 3}\right)=\ln x. \]

## 17.6 Practice

**Problem 17.1 **
Write each equation into equivalent exponential form.

- \(\log_37=y\)
- \(3=\log_b64\)
- \(\log x=y\)
- \(\ln(x-1)=c\)

**Problem 17.2 **
Write each equation into equivalent logarithmic form.

- \(7^x=10\)
- \(b^5=2\)
- \(e^{2y-1}=x\)
- \(10^x=c^2+1\)

**Problem 17.3 **
Evaluate.

- \(\log_216\)
- \(\log_93\)
- \(\log 10\)
- \(\ln 1\)

**Problem 17.4 **
Evaluate.

- \(e^{\ln 2}\)
- \(\log 10^{\frac13}\)
- \(\ln(\sqrt{e})\)
- \(\log_2(\frac12)\)

**Problem 17.5 **
Find the domain of the function \(f(x)=\log(x-5)\). Write in interval notation.

**Problem 17.6 **
Sketch the graph of each function and find its range.

- \(f(x)=\log_2x\)
- \(f(x)=\log_{\frac12} x\)

**Problem 17.7 **
Expand the logarithm and simplify.

- \(\log(100x)\)
- \(\ln\left(\frac{10}{e^2}\right)\)
- \(\log_b(\sqrt[3]{x})\)
- \(\log_7(\frac{x^2\sqrt{y}}{z})\)

**Problem 17.8 **
Expand the logarithm and simplify.

- \(\log_b\sqrt{\frac{x^2y}{5}}\)
- \(\ln(\sqrt[3]{(x^2+1)y^{-2}})\)
- \(\log(x\sqrt{10x}-\sqrt{10x})\)

**Problem 17.9 **
Write as a single logarithm.

- \(\frac13\log x +\log y\)
- \(\frac12\ln(x^2+1)-2\ln x\)
- \(\frac13\log_2 x - 3\log_2(x+1)+1\)

**Problem 17.10 **
Write as a single logarithm.

- \(2\log(2x+1)-\frac12\log x\)
- \(3\ln x - 5\ln y + \frac{1}{2}\ln z\)
- \(3\log_3 x-2\log_3(1-x)+\frac13\log_3 (x^2+1)\).

**Problem 17.11 **
Evaluate the logarithm and round it to the nearest hundredth.

- \(\log_2 10\)
- \(\log_3 5\)
- \(\dfrac{1}{\log_52}\)
- \(\log_45-\log_29\)

**Problem 17.12 **
Simplify the logarithmic expression
\[
\frac{\log_3(x^2)\log_y\sqrt{3}}{\log x}.
\]