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?

Yes — all study notes, flashcards, and practice problems for Logarithmic Functions on Study Mondo are free to access. No account is needed.

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.

Logarithmic Functions

What is a Logarithm?

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

Exponential Form	Logarithmic Form
$2^3 = 8$	$\log_2(8) = 3$
$10^2 = 100$	$\log(100) = 2$
$e^x = 5$	$\ln(5) = x$
$b^y = x$	$\log_b(x) = y$

❓ Question:

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

Solution:

Part a) $\log_3(27)$

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

$3^? = 27$

❓ Question:

Evaluate the following logarithms without a calculator:

a) $\log_2 32$ b) $\log_5 \frac{1}{25}$ c)

❓ Question:

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

❓ Question:

Use properties of logarithms to expand the following expression completely:

$\log_3 \left(\frac{x^4\sqrt{y}}{z^2}\right)$

❓ Question:

Solve for $x$ : $\log_4(x) = 3$

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

Solution:

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

$2^5 = 32$

Therefore: $\log_2 32 = 5$

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

$\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$

Therefore: $\log_5 \frac{1}{25} = -2$

Part (c): $\ln e^7$ means $\log_e e^7$

By the property $\log_b b^x = x$ :

$\ln e^7 = 7$

Solution:

Step 1: Apply the quotient rule.

$\log_2\left(\frac{8x^3}{y^2}\right) = \log_2(8x^3) - \log_2(y^2)$

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

$= \log_2(8) + \log_2(x^3) - \log_2(y^2)$

Step 3: Apply the power rule.

$= \log_2(8) + 3\log_2(x) - 2\log_2(y)$

Step 4: Simplify $\log_2(8)$ .

Since $2^3 = 8$ , we have $\log_2(8) = 3$

$= 3 + 3\log_2(x) - 2\log_2(y)$

Answer: $3 + 3\log_2(x) - 2\log_2(y)$

Solution:

We'll use three key properties:

Product rule: $\log_b(MN) = \log_b M + \log_b N$
Quotient rule: $\log_b(M/N) = \log_b M - \log_b N$
Power rule: $\log_b M^p = p\log_b M$

Step 1: Apply quotient rule:

$\log_3 \left(\frac{x^4\sqrt{y}}{z^2}\right) = \log_3(x^4\sqrt{y}) - \log_3(z^2)$

Step 2: Apply product rule to the first term:

$= \log_3(x^4) + \log_3(\sqrt{y}) - \log_3(z^2)$

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

$= 4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z$

Final answer: $4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z$

Solution:

Step 1: Convert from logarithmic form to exponential form.

$\log_4(x) = 3 \quad \Rightarrow \quad 4^3 = x$

Step 2: Evaluate.

$x = 4^3 = 64$

Step 3: Check (optional but recommended).

$\log_4(64) = \log_4(4^3) = 3$ ✓

Answer: $x = 64$

Logarithmic Functions

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Logarithmic Functions

What is a Logarithm?

Definition

Common Logarithms

Converting Between Forms

Properties of Logarithms

Product Rule

Quotient Rule

Power Rule

Change of Base Formula

Special Logarithm Values