Exponent and Log Rules

Why We Rewrite Expressions

In Calculus, you will often start with a fraction like one over the square root of x. To get it ready for the Power Rule, we first identify the root as x to the power of one-half. Then, we move it to the top by changing the sign. Now, x to the negative one-half is ready for Calculus.

The Rules of Flattening

To flatten any expression, use these two rules. For roots, remember that the root index—the number on the shelf—always becomes the denominator. For fractions, a negative sign acts like a ticket to move a variable across the fraction bar. Here, the cube root of x squared becomes x to the two-thirds power. And here, one over x cubed becomes x to the negative three.

Logarithm Expansion

Natural logs appear everywhere in Calculus. The log Power Rule is your best friend; it lets you pull an exponent down to the front as a multiplier. Also, remember that 'ln' and 'e' are inverses—they simply undo each other, leaving you with just what was inside.

Practice: Rewriting Roots

Try rewriting this expression. First, convert the root in the denominator into a fractional exponent. Correct. The fourth root of x cubed becomes x to the three-fourths. Now, move it to the numerator. Perfect. Moving it up changes the sign, giving us five times x to the negative three-fourths.

Practice: Expanding Logs

Expand this log as much as possible. Start by applying the Quotient Rule to split the fraction. Exactly. The 3 moves to the front, and the ln and e cancel out, leaving us with three ln x minus x. Good. Now we have ln of x cubed minus ln of e to the x. Next, apply the log Power Rule and the Undo Rule.

Mixed Review: Signs & Exponents

Simplify this expression: negative 2 x to the negative 2, times negative 3 x to the 5. Start with the coefficients. Negative 2 plus 5 is 3. The final simplified form is 6 x cubed. Negative 2 times negative 3 is positive 6. Now, both bases are x, and when you multiply the same base you add the exponents: x to the a times x to the b is x to the a plus b.

Calculus-Ready Challenge

Great work. This expression is now ready for Calculus. Doing the algebra first makes the derivative a one-line Power Rule step. Here is a typical Calculus setup. Rewrite each of these three terms into a 'flat' format. Click each term to transform it, then we will see the final result. 4 over x squared becomes 4 x to the negative 2. The fifth root of x becomes x to the one-fifth. And ln of e to the 2x simply becomes 2x.

Takeaways: Exponents & Logs

Let's recap. Roots become fractional exponents, and one over a power becomes a negative exponent. Logs allow us to simplify complex powers, and the natural log and e-constant cancel each other out. Hover over the terms to lock in what you just practiced.