Grade 11

Advanced

6 min

Lesson 4 of 12

Logarithmic Foundations

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Understand logarithms as the inverse of exponentiation and learn to switch between forms.

How do scientists measure the massive energy of an earthquake or the microscopic acidity of a lemon on the same scale? They use logarithms—the mathematical 'magnifying glass' that turns multiplication into addition and tames the infinite.

Learning Objectives

1.By the end of this lesson, you will be able to define a logarithm as the inverse of an exponential function.
2.You'll understand how to convert equations between exponential form $b^y = x$ and logarithmic form $\log_b(x) = y$ .
3.You will be able to identify the domain and range of logarithmic functions and evaluate natural logarithms.

The Inverse Relationship

At its core, a logarithm is simply an exponent. If we know that

2^3 = 8

, the logarithm asks the reverse question: 'To what power must we raise 2 to get 8?' The answer is 3. This is written as

\log_2(8) = 3

. In general, the relationship is defined as:

b^y = x \iff \log_b(x) = y

where

b

is the base,

y

is the exponent (or the logarithm), and

x

is the argument. Because they are inverses, the logarithmic function 'undoes' the exponential function. This relationship is the key to solving equations where the variable is trapped in the exponent.

Convert the following from exponential to logarithmic form: 1. Given $5^2 = 25$ , identify the base ( $b=5$ ), the exponent ( $y=2$ ), and the result ( $x=25$ ). 2. Rewrite using the log definition: $\log_5(25) = 2$ . 3. Given $10^{-2} = 0.01$ , rewrite as $\log_{10}(0.01) = -2$ .

Quick Check

What is the value of $\log_3(81)$ ?

Ask yourself: 3 raised to what power equals 81?

Answer

Common and Natural Logarithms

In mathematics and science, two bases are so common they have their own notation. The Common Logarithm has a base of 10 and is usually written simply as $\log(x)$ without the subscript. If you see no base, assume it is 10. The Natural Logarithm has the irrational base $e \approx 2.718$ and is written as $\ln(x)$ . The natural log is the 'language of growth' in calculus and biology. For both, the same rules apply: $\ln(x) = y$ is equivalent to $e^y = x$ .

Evaluate the following without a calculator: 1. $\log(1000)$ : Since the base is 10, we solve $10^y = 1000$ . $10^3 = 1000$ , so $\log(1000) = 3$ . 2. $\ln(e^5)$ : Since the base is $e$ , we solve $e^y = e^5$ . Clearly, $y = 5$ . 3. $\log(\sqrt{10})$ : Rewrite the square root as an exponent: $\log(10^{1/2})$ . The answer is $1/2$ .

Quick Check

True or False: $\ln(1) = 0$ .

Any base raised to the power of 0 equals 1.

Answer

True

The Domain Restriction

Because logarithms are the inverse of exponentials, their domain and range are swapped. An exponential function $f(x) = b^x$ always produces a positive result ( $x > 0$ ). Therefore, you can only take the logarithm of a positive number. The domain of $f(x) = \log_b(x)$ is $(0, \infty)$ . You cannot take the log of zero or a negative number in the real number system. Conversely, the range of a logarithm is $(-\infty, \infty)$ , meaning a logarithm can result in any real number (negative, positive, or zero).

Find the domain of the function $f(x) = \log_2(x - 5)$ . 1. Set the argument to be greater than zero: $x - 5 > 0$ . 2. Solve the inequality: $x > 5$ . 3. Express in interval notation: The domain is $(5, \infty)$ . The graph has a vertical asymptote at $x = 5$ .

Key Takeaways

1A logarithm is the inverse of an exponent: $\log_b(x) = y$ means $b^y = x$ .
2The 'Common Log' ( $\\log$ ) uses base 10, while the 'Natural Log' ( $\\ln$ ) uses base $e$ .
3The domain of a logarithmic function is restricted to positive numbers ( $x > 0$ ).
4Logarithms can output any real number, including negative values.

Quick Check

Multiple Choice

Which of the following is the logarithmic form of $4^x = 64$ ?

Multiple Choice

What is the value of $\log_5(\frac{1}{25})$ ?

True/False

The expression $\ln(-2)$ is a defined real number.

What's Next?

Review Tomorrow

In 24 hours, try to explain to someone else why $\log_b(1) = 0$ for any valid base $b$ .

Practice Activity

Look up the pH scale or the Richter scale and identify which base they use for their logarithmic calculations.

Browse

Browse

Logarithmic Foundations

Learning Objectives

The Inverse Relationship

Common and Natural Logarithms

The Domain Restriction

Key Takeaways

Quick Check

What's Next?

Logarithmic Foundations

Learning Objectives

The Inverse Relationship

Common and Natural Logarithms

The Domain Restriction

Key Takeaways

Quick Check

What's Next?