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Logarithmic Functions

Understanding logarithms as inverse of exponentials, logarithmic properties, and solving logarithmic equations

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Logarithmic Functions

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. It answers the question:

"To what power must we raise the base to get a certain number?"

Definition

log⁡b(x)=ymeansby=x\log_b(x) = y \quad \text{means} \quad b^y = x

Read as: "log base bb of xx equals yy"

Example: log⁡2(8)=3\log_2(8) = 3 because 23=82^3 = 8

Common Logarithms

  1. Common logarithm (base 10): log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x)
  2. Natural logarithm (base ee): ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x)

Converting Between Forms

Exponential FormLogarithmic Form
23=82^3 = 8log⁡2(8)=3\log_2(8) = 3
102=10010^2 = 100log⁡(100)=2\log(100) = 2
ex=5e^x = 5ln⁡(5)=x\ln(5) = x
by=xb^y = xlog⁡b(x)=y\log_b(x) = y

Properties of Logarithms

Product Rule

log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)

The log of a product is the sum of the logs.

Quotient Rule

log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

The log of a quotient is the difference of the logs.

Power Rule

log⁡b(Mp)=p⋅log⁡b(M)\log_b(M^p) = p \cdot \log_b(M)

The log of a power is the exponent times the log.

Change of Base Formula

log⁡b(x)=log⁡a(x)log⁡a(b)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)}

Useful for calculating logs with different bases on a calculator.

Special Logarithm Values

  1. log⁡b(1)=0\log_b(1) = 0 because b0=1b^0 = 1
  2. log⁡b(b)=1\log_b(b) = 1 because b1=bb^1 = b
  3. log⁡b(bx)=x\log_b(b^x) = x (inverse property)
  4. blog⁡b(x)=xb^{\log_b(x)} = x (inverse property)

Logarithmic Functions

The function f(x)=log⁡b(x)f(x) = \log_b(x) has these properties:

  • Domain: (0,∞)(0, \infty) (only positive numbers)
  • Range: (−∞,∞)(-\infty, \infty) (all real numbers)
  • Vertical Asymptote: x=0x = 0 (the y-axis)
  • x-intercept: (1,0)(1, 0) since log⁡b(1)=0\log_b(1) = 0
  • Always increasing if b>1b > 1
  • Always decreasing if 0<b<10 < b < 1

Solving Logarithmic Equations

Strategy 1: Convert to exponential form

Strategy 2: Use logarithm properties to combine/simplify

Strategy 3: Check your answers (domain restrictions!)

Important Note

⚠️ You cannot take the log of a negative number or zero!

The domain of log⁡b(x)\log_b(x) requires x>0x > 0.

📚 Practice Problems

1Problem 1easy

❓ Question:

Evaluate the following logarithms: (a) log⁡3(27)\log_3(27), (b) log⁡5(125)\log_5(\frac{1}{25}), (c) log⁡(10000)\log(10000)

💡 Show Solution

Solution:

Part a) log⁡3(27)\log_3(27)

Ask: "3 raised to what power equals 27?"

3?=273^? = 27 33=273^3 = 27

Therefore: log⁡3(27)=3\log_3(27) = 3

Part b) log⁡5(125)\log_5\left(\frac{1}{25}\right)

Ask: "5 raised to what power equals 125\frac{1}{25}?"

5?=1255^? = \frac{1}{25}

We know 52=255^2 = 25, so 5−2=1255^{-2} = \frac{1}{25}

Therefore: log⁡5(125)=−2\log_5\left(\frac{1}{25}\right) = -2

Part c) log⁡(10000)\log(10000) (base 10)

Ask: "10 raised to what power equals 10000?"

10?=1000010^? = 10000 104=1000010^4 = 10000

Therefore: log⁡(10000)=4\log(10000) = 4

Answers: (a) 3, (b) -2, (c) 4

2Problem 2easy

❓ Question:

Evaluate the following logarithms without a calculator:

a) log⁡232\log_2 32 b) log⁡5125\log_5 \frac{1}{25} c) ln⁡e7\ln e^7

💡 Show Solution

Solution:

Part (a): log⁡232\log_2 32 means "2 to what power equals 32?"

25=322^5 = 32

Therefore: log⁡232=5\log_2 32 = 5

Part (b): log⁡5125\log_5 \frac{1}{25} means "5 to what power equals 125\frac{1}{25}?"

125=152=5−2\frac{1}{25} = \frac{1}{5^2} = 5^{-2}

Therefore: log⁡5125=−2\log_5 \frac{1}{25} = -2

Part (c): ln⁡e7\ln e^7 means log⁡ee7\log_e e^7

By the property log⁡bbx=x\log_b b^x = x:

ln⁡e7=7\ln e^7 = 7

3Problem 3medium

❓ Question:

Expand using logarithm properties: log⁡2(8x3y2)\log_2\left(\frac{8x^3}{y^2}\right)

💡 Show Solution

Solution:

Step 1: Apply the quotient rule.

log⁡2(8x3y2)=log⁡2(8x3)−log⁡2(y2)\log_2\left(\frac{8x^3}{y^2}\right) = \log_2(8x^3) - \log_2(y^2)

Step 2: Apply the product rule to the first term.

=log⁡2(8)+log⁡2(x3)−log⁡2(y2)= \log_2(8) + \log_2(x^3) - \log_2(y^2)

Step 3: Apply the power rule.

=log⁡2(8)+3log⁡2(x)−2log⁡2(y)= \log_2(8) + 3\log_2(x) - 2\log_2(y)

Step 4: Simplify log⁡2(8)\log_2(8).

Since 23=82^3 = 8, we have log⁡2(8)=3\log_2(8) = 3

=3+3log⁡2(x)−2log⁡2(y)= 3 + 3\log_2(x) - 2\log_2(y)

Answer: 3+3log⁡2(x)−2log⁡2(y)3 + 3\log_2(x) - 2\log_2(y)

4Problem 4medium

❓ Question:

Use properties of logarithms to expand the following expression completely:

log⁡3(x4yz2)\log_3 \left(\frac{x^4\sqrt{y}}{z^2}\right)

💡 Show Solution

Solution:

We'll use three key properties:

  • Product rule: log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N
  • Quotient rule: log⁡b(M/N)=log⁡bM−log⁡bN\log_b(M/N) = \log_b M - \log_b N
  • Power rule: log⁡bMp=plog⁡bM\log_b M^p = p\log_b M

Step 1: Apply quotient rule:

log⁡3(x4yz2)=log⁡3(x4y)−log⁡3(z2)\log_3 \left(\frac{x^4\sqrt{y}}{z^2}\right) = \log_3(x^4\sqrt{y}) - \log_3(z^2)

Step 2: Apply product rule to the first term:

=log⁡3(x4)+log⁡3(y)−log⁡3(z2)= \log_3(x^4) + \log_3(\sqrt{y}) - \log_3(z^2)

Step 3: Rewrite y=y1/2\sqrt{y} = y^{1/2} and apply power rule:

=4log⁡3x+12log⁡3y−2log⁡3z= 4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z

Final answer: 4log⁡3x+12log⁡3y−2log⁡3z4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z

5Problem 5easy

❓ Question:

Solve for xx: log⁡4(x)=3\log_4(x) = 3

💡 Show Solution

Solution:

Step 1: Convert from logarithmic form to exponential form.

log⁡4(x)=3⇒43=x\log_4(x) = 3 \quad \Rightarrow \quad 4^3 = x

Step 2: Evaluate.

x=43=64x = 4^3 = 64

Step 3: Check (optional but recommended).

log⁡4(64)=log⁡4(43)=3\log_4(64) = \log_4(4^3) = 3 ✓

Answer: x=64x = 64

Explain using:

⚠️ Common Mistakes: Logarithmic Functions

Avoid these 4 frequent errors

🌍 Real-World Applications: Logarithmic Functions

See how this math is used in the real world

📝 Worked Example: Related Rates — Expanding Circle

Problem:

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of 22 cm/s. How fast is the area of the circle increasing when the radius is 1010 cm?

2Write the relationship between variables
3Differentiate both sides with respect to time
4Substitute known values
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Solving Exponential and Logarithmic Equations
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📌 Related Topics in Exponential and Logarithmic Functions

Exponential Functions and ModelingLogarithmic Functions and EquationsExponential Functions and Growth/DecaySolving Exponential and Logarithmic EquationsLogarithmic and Exponential Models

❓ Frequently Asked Questions

What is Logarithmic Functions?▾
Understanding logarithms as inverse of exponentials, logarithmic properties, and solving logarithmic equations
How can I study Logarithmic Functions effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Logarithmic Functions study guide free?▾
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What course covers Logarithmic Functions?▾
Logarithmic Functions is part of the AP Precalculus course on Study Mondo, specifically in the Exponential and Logarithmic Functions section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Logarithmic Functions?▾
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.