Exponent Laws and Logarithmic Properties

There are several rules that are helpful when working with exponential functions.

Law of Exponents:

\(a^xa^y = a^{x+y}\)
\((a^x)^y = a^{xy}\)
\(\left(\cfrac{a}{b}\right)^x = \cfrac{a^x}{b^x}\)
\(\cfrac{a^x}{a^y} = a^{x-y}\)
\((ab)^x = a^xb^x\)

The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents. The third law states that in order to raise a power to a new power, we multiply the exponents. The fourth and fifth laws state that in order to raise a product or quotient to a power, we raise each factor to that power.

Example: Simplify the expression:

\(\cfrac{7^{3x}\cdot7^{x+1}}{7^{2x+5}}\)
\(=7^{3x+(x+1)-(2x+5)}\)
\(=7^{3x+x+1-2x-5}\)
\(=7^{2x-4}\)

Example: Simplify the expression:

\(\left(\cfrac{3^{4x}}{2^x}\right)^3\)
\(=\cfrac{(3^{4x})^3}{(2^x)^3}\)
\(=\cfrac{3^{12x}}{2^{3x}}\)

Note: We cannot simplify any further since the terms in the numerator and denominator do not have the same base.

Log Laws

There are three properties that are useful when working with logarithmic functions.

Properties of Logarithms

If x, y > 0 and r is any real number, then

log_a(xy) = log_a x + log_ay

log_a(x/y) = log_a x - log_ay

log_a(x^r) = rlog_ax

Example: Simplify the expression log₂5 + log₂3

Solution:

log₂5 + log₂3

= log₂(5 x 3)

= log₂(15)

Example: Simplify the expression 5(In2)

5(In2)

= In(2⁵)

= In(32)

Exponent Laws Example: