Learn on PengienVision, Algebra 2Chapter 6: Exponential and Logarithmic Functions

Lesson 3: Logarithms

In this Grade 11 enVision Algebra 2 lesson, students learn what logarithms are and how to evaluate them by understanding that a logarithm is the inverse of exponentiation. Students practice converting between exponential form and logarithmic form, evaluating expressions such as log base 5 of 125, and identifying when a logarithmic expression is undefined. The lesson also introduces common logarithms and natural logarithms as part of Chapter 6 on Exponential and Logarithmic Functions.

Section 1

Definition of Logarithm

Property

For $b > 0$ , $b \neq 1$ , the base $b$ logarithm of $x$ , written $\log_b x$ , is the exponent to which $b$ must be raised in order to yield $x$ . This gives the following conversion equations:

y = \log_b x \quad \text{if and only if} \quad x = b^y

Examples

The logarithmic statement $\log_3 81 = 4$ is the same as asking "what power of 3 equals 81?", which is answered by the exponential statement $3^4 = 81$ .
To solve the equation $10^x = 100$ , you can rewrite it in logarithmic form as $x = \log_{10} 100$ , which means $x = 2$ .
The expression $\log_7 1$ asks "7 to what power equals 1?" Since any number to the power of 0 is 1, $\log_7 1 = 0$ .

Explanation

A logarithm answers the question: "What exponent do I need?" It undoes an exponential function, allowing you to solve for a variable that is stuck in the exponent. It's the key to unlocking those tricky exponential equations.

Section 2

Evaluate Logarithmic Functions

Property

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation. To find the exact value of a logarithm like $\log_b N$ , you can set the expression equal to $x$ (so $\log_b N = x$ ) and then convert it into the exponential equation $b^x = N$ to solve for $x$ .

Examples

To find the value of $x$ in $\log_x 81 = 2$ , convert to exponential form: $x^2 = 81$ . Since the base must be positive, $x=9$ .
To find the value of $x$ in $\log_2 x = 5$ , convert to exponential form: $2^5 = x$ . Therefore, $x=32$ .
To find the exact value of $\log_5 \frac{1}{25}$ , set it to $x$ . $\log_5 \frac{1}{25} = x$ becomes $5^x = \frac{1}{25}$ . Since $\frac{1}{25} = 5^{-2}$ , we find $x = -2$ .

Explanation

Evaluating a logarithm means finding the exponent. By asking, 'What power do I raise the base to get the number?', you are setting up the problem. Converting to an exponential equation is the formal way to solve for that unknown exponent.

Section 3

Base 10 logarithms

Property

Base 10 logarithms are called common logarithms.

The subscript 10 is often omitted, so that $\log x$ is understood to mean $\log_{10} x$ . To evaluate a base 10 logarithm, we use the LOG key on a calculator.

Examples

To find $\log_{10} 1000$ , we know $10^3=1000$ , so $\log 1000 = 3$ .
To approximate $\log 350$ , use a calculator: $\log 350 \approx 2.5441$ . This means $10^{2.5441} \approx 350$ .
To solve $10^x = 12$ , we rewrite it as $x = \log 12$ . Using a calculator, $x \approx 1.0792$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Definition of Logarithm

Property

For $b > 0$ , $b \neq 1$ , the base $b$ logarithm of $x$ , written $\log_b x$ , is the exponent to which $b$ must be raised in order to yield $x$ . This gives the following conversion equations:

y = \log_b x \quad \text{if and only if} \quad x = b^y

Examples

The logarithmic statement $\log_3 81 = 4$ is the same as asking "what power of 3 equals 81?", which is answered by the exponential statement $3^4 = 81$ .
To solve the equation $10^x = 100$ , you can rewrite it in logarithmic form as $x = \log_{10} 100$ , which means $x = 2$ .
The expression $\log_7 1$ asks "7 to what power equals 1?" Since any number to the power of 0 is 1, $\log_7 1 = 0$ .

Explanation

Section 2

Evaluate Logarithmic Functions

Property

Examples

To find the value of $x$ in $\log_x 81 = 2$ , convert to exponential form: $x^2 = 81$ . Since the base must be positive, $x=9$ .
To find the value of $x$ in $\log_2 x = 5$ , convert to exponential form: $2^5 = x$ . Therefore, $x=32$ .
To find the exact value of $\log_5 \frac{1}{25}$ , set it to $x$ . $\log_5 \frac{1}{25} = x$ becomes $5^x = \frac{1}{25}$ . Since $\frac{1}{25} = 5^{-2}$ , we find $x = -2$ .

Explanation

Section 3

Base 10 logarithms

Property

Base 10 logarithms are called common logarithms.

The subscript 10 is often omitted, so that $\log x$ is understood to mean $\log_{10} x$ . To evaluate a base 10 logarithm, we use the LOG key on a calculator.

Examples

To find $\log_{10} 1000$ , we know $10^3=1000$ , so $\log 1000 = 3$ .
To approximate $\log 350$ , use a calculator: $\log 350 \approx 2.5441$ . This means $10^{2.5441} \approx 350$ .
To solve $10^x = 12$ , we rewrite it as $x = \log 12$ . Using a calculator, $x \approx 1.0792$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Definition of Logarithm

Property

For $b > 0$ , $b \neq 1$ , the base $b$ logarithm of $x$ , written $\log_b x$ , is the exponent to which $b$ must be raised in order to yield $x$ . This gives the following conversion equations:

y = \log_b x \quad \text{if and only if} \quad x = b^y

Examples

The logarithmic statement $\log_3 81 = 4$ is the same as asking "what power of 3 equals 81?", which is answered by the exponential statement $3^4 = 81$ .
To solve the equation $10^x = 100$ , you can rewrite it in logarithmic form as $x = \log_{10} 100$ , which means $x = 2$ .
The expression $\log_7 1$ asks "7 to what power equals 1?" Since any number to the power of 0 is 1, $\log_7 1 = 0$ .

Explanation

Section 2

Evaluate Logarithmic Functions

Property

Examples

To find the value of $x$ in $\log_x 81 = 2$ , convert to exponential form: $x^2 = 81$ . Since the base must be positive, $x=9$ .
To find the value of $x$ in $\log_2 x = 5$ , convert to exponential form: $2^5 = x$ . Therefore, $x=32$ .
To find the exact value of $\log_5 \frac{1}{25}$ , set it to $x$ . $\log_5 \frac{1}{25} = x$ becomes $5^x = \frac{1}{25}$ . Since $\frac{1}{25} = 5^{-2}$ , we find $x = -2$ .

Explanation

Section 3

Base 10 logarithms

Property

Base 10 logarithms are called common logarithms.

The subscript 10 is often omitted, so that $\log x$ is understood to mean $\log_{10} x$ . To evaluate a base 10 logarithm, we use the LOG key on a calculator.

Examples

To find $\log_{10} 1000$ , we know $10^3=1000$ , so $\log 1000 = 3$ .
To approximate $\log 350$ , use a calculator: $\log 350 \approx 2.5441$ . This means $10^{2.5441} \approx 350$ .
To solve $10^x = 12$ , we rewrite it as $x = \log 12$ . Using a calculator, $x \approx 1.0792$ .