Exponent & Logarithm Rules for Calculus

Calculus requires fluent translation between radical forms ($\sqrt{x}$) and exponential forms ($x^{1/2}$), as well as solving exponential and logarithmic equations.

Core Theorem

Key Properties:

a^{-x} = \frac{1}{a^x}

a^{p/q} = \sqrt[q]{a^p}

, and

ln(ab) = ln(a) + ln(b)

. example

Practice Exercises

Example 01Easy

Rewrite using exponents and solve:

2x(4-x)^{-1/2} - 3\sqrt{4-x} = 0

NEED A HINT?

Convert

\sqrt{4-x}

(4-x)^{1/2}

. Factor out the term with the smallest exponent:

(4-x)^{-1/2}

. example

SHOW DETAILED EXPLANATION

Step 1: Convert to Exponents

Rewrite the equation:

2x(4-x)^{-1/2} - 3(4-x)^{1/2} = 0

Step 2: Factor Out Common Term

Factor out

(4-x)^{-1/2}

(4-x)^{-1/2} [ 2x - 3(4-x)^1 ] = 0

. Note that subtracting exponents:

1/2 - (-1/2) = 1

Step 3: Solve Linear Part

Simplify inside the bracket:

2x - 12 + 3x = 5x - 12 = 0 \Rightarrow x = 12/5

. (Check domain:

4-x > 0

Example 02Easy

Solve the logarithmic inequality:

ln(x^2 - 2x - 2) \le 0

NEED A HINT?

Convert the log form to exponential form (

e^0=1

) AND ensure the argument of the log is positive (

x^2-2x-2 > 0

). example

SHOW DETAILED EXPLANATION

Step 1: Remove Log

x^2 - 2x - 2 \le e^0 \Rightarrow x^2 - 2x - 2 \le 1 \Rightarrow x^2 - 2x - 3 \le 0

Step 2: Solve Outer Inequality

Factor

(x-3)(x+1) \le 0

. Solution:

[-1, 3]

Step 3: Check Domain

You must also satisfy

x^2 - 2x - 2 > 0

. Find roots

1 \pm \sqrt{3}

. The valid

x

must be outside

(1-\sqrt{3}, 1+sqrt{3})

. Intersect this with

[-1, 3]

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