Exponents & Logarithms

Master the inverse relationship between exponential and logarithmic functions, and learn to convert fluently between the two forms.

Algebra 2

What Is an Exponent?

An exponential expression \( b^x = y \) tells us to multiply the base \( b \) by itself \( x \) times to get the result \( y \).

Base \( b \)

The number being multiplied repeatedly. Must be positive and \( b \neq 1 \).

Exponent \( x \)

How many times the base is used as a factor (the "power").

Result \( y \)

The output value produced by the exponentiation.

What Is a Logarithm?

A logarithm answers the question: "What exponent do I need?" Specifically, \( \log_b y = x \) asks: "To what power must I raise \( b \) to get \( y \)?"

Base \( b \)

Same base as the corresponding exponential. Written as a subscript.

Argument \( y \)

The number you're taking the log of. Must be positive: \( y > 0 \).

Output \( x \)

The exponent that the base needs: the answer to the log.

Why Do Logarithms Matter?

Logarithms let us undo exponentiation, just like subtraction undoes addition. They appear everywhere in science, finance, and technology.

Earthquake Scales

The Richter scale uses \( \log_{10} \) to measure seismic intensity.

Compound Interest

Logs help solve "how long until my investment doubles?"

Computer Science

Algorithm complexity often involves \( \log_2 n \) (binary search, etc.).

The Inverse Relationship

Exponential and logarithmic functions are inverses of each other. This means they "undo" one another:

Graphs: Mirror Images

Because they are inverses, the graphs of \( y = b^x \) and \( y = \log_b x \) are reflections across the line \( y = x \).

Exponential vs. Logarithmic Form

	Exponential Form	Logarithmic Form
General	\( b^x = y \)	\( \log_b y = x \)
Example 1	\( 3^4 = 81 \)	\( \log_3 81 = 4 \)
Example 2	\( 10^{-2} = 0.01 \)	\( \log_{10} 0.01 = -2 \)
Example 3	\( e^1 = e \)	\( \ln e = 1 \)
Solves for	The result \( y \)	The exponent \( x \)

The Two Cancellation Identities

Since exponents and logarithms are inverses, composing them in either order returns the original input:

Identity 1

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

Take the log of an exponential: get the exponent back.

Identity 2

\( b^{\log_b x} = x \)

Raise the base to a log: get the argument back.

Special Bases to Know

Common Log

Base 10: \( \log x = \log_{10} x \). Used in pH, decibels, and the Richter scale.

Natural Log

Base \( e \approx 2.718 \): \( \ln x = \log_e x \). Used in calculus, continuous growth, and physics.

Binary Log

Base 2: \( \log_2 x \). Fundamental in computer science and information theory.

How to Convert: Step by Step

1

Identify the three parts: the base \( b \), the exponent \( x \), and the result \( y \).

2

Exponential → Log: rewrite \( b^x = y \) as \( \log_b y = x \). The base stays, the result becomes the argument.

3

Log → Exponential: rewrite \( \log_b y = x \) as \( b^x = y \). The base is raised to the output.

4

Verify: Check by substituting back: does \( b^x \) actually equal \( y \)?

Example 1: Convert to Log Form

Convert \( 5^3 = 125 \) to logarithmic form.

1

Identify: base \( b = \) 5, exponent \( x = \) 3, result \( y = \) 125.

2

Apply the rule: \( b^x = y \longrightarrow \log_b y = x \).

3

Write: \( \log_5 125 = 3 \).

Example 2: Convert to Exponential Form

Convert \( \log_4 64 = 3 \) to exponential form.

1

Identify: base \( b = \) 4, argument \( y = \) 64, output \( x = \) 3.

2

Apply: \( \log_b y = x \longrightarrow b^x = y \).

3

Write: \( 4^3 = 64 \).

Example 3: Solving with Logs

Solve for \( x \): \( 3^x = 243 \).

1

Convert to log form: \( x = \log_3 243 \).

2

Evaluate: Express 243 as a power of 3: \( 3^5 = 243 \).

3

Apply the cancellation identity: \( \log_3(3^5) = 5 \).

4

Answer: \( x = 5 \).

Example 4: Simplify with Identities

Simplify each expression:

a) \( \log_7(7^{11}) \)

By Identity 1: \( \log_b(b^x) = x \), so the answer is 11.

b) \( 10^{\log_{10} 50} \)

By Identity 2: \( b^{\log_b x} = x \), so the answer is 50.

Common Mistakes

✕ Swapping Base and Argument

Writing \( \log_{81} 3 = 4 \) instead of \( \log_3 81 = 4 \). The base stays as the base in both forms.

✕ Log of Negatives

Trying to compute \( \log_2(-8) \). Logarithms are only defined for positive arguments: \( y > 0 \).

✕ Treating Log as Multiplication

Writing \( \log(a + b) = \log a + \log b \). This is false. The product rule is \( \log(ab) = \log a + \log b \).

Summary Reference Table

Property	Exponential	Logarithmic
Form	\( y = b^x \)	\( y = \log_b x \)
Solves for	The result \( y \)	The exponent \( y \)
Domain	\( x \in \mathbb{R} \)	\( x > 0 \)
Range	\( y > 0 \)	\( y \in \mathbb{R} \)
Identity	\( b^{\log_b x} = x \)	\( \log_b(b^x) = x \)
Graph passes through	\( (0, 1) \)	\( (1, 0) \)
Asymptote	\( y = 0 \) (horizontal)	\( x = 0 \) (vertical)

Question 1 of 5

Which is the logarithmic form of \( 2^7 = 128 \)?

The base of the exponential (2) becomes the base of the logarithm, the result (128) becomes the argument, and the exponent (7) becomes the output: \( \log_2 128 = 7 \).

Question 2 of 5

Simplify: \( \log_5(5^9) \)

By the cancellation identity \( \log_b(b^x) = x \): the base of the log (5) matches the base of the power (5), so they cancel, leaving the exponent: 9.

Question 3 of 5

Solve for \( x \): \( \log_3 x = 4 \)

Convert to exponential form: \( 3^4 = x \). Calculate: \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \). So \( x = 81 \).

Question 4 of 5

What is the value of \( 10^{\log_{10} 42} \)?

By the cancellation identity \( b^{\log_b x} = x \): the base 10 and the \( \log_{10} \) cancel each other, leaving the argument: 42.

Question 5 of 5

Which of the following is the exponential form of \( \ln 1 = 0 \)?

Since \( \ln \) means \( \log_e \), we have \( \log_e 1 = 0 \). Converting: the base \( e \) raised to the output \( 0 \) equals the argument \( 1 \): \( e^0 = 1 \).