Laws of Exponents
Properties of Exponents
The laws of exponents are as follows:
- Product Rule: $a^x \cdot a^y = a^{x+y}$
- Quotient Rule: $\frac{a^x}{a^y} = a^{x-y}$
- Power Rule: $(a^x)^y = a^{xy}$
These rules apply regardless of whether $x$ and $y$ are integers, rational, or real numbers.
Example- $(4^2)^5 = 4^{10}$
- $4^6 \times 4^8 = 4^{14}$
- $\frac{5^4}{5^2}=5^2$
$x^{y^z}$ is not the same as $(x^y)^z$
- The first one is the power being applied on $y$, for example
$$2^{2^3}=2^{2\times 2\times 2}=2^8=256$$
- The second is is the power being applied to the entire number $x^y$ for example
$$(2^3)^2 = 2^6 = 64$$
Negative Exponents
if you have $x^{-a}$ for any $a$
$$x^{-a} = \frac{1}{x^a}$$
Example- $2^{-2} = \frac{1}{2^2}=\frac{1}{4}$
- $\frac{1}{3^{-3}}=3^3=27$
- $5^2\times(5^2)^{-2}=5^2\times5^{-4}=5^{-2}=\frac{1}{5^2}=\frac{1}{25}$
Logarithms: An Introduction
Logarithms are one of the inverse functions to exponentiation, the other being radicals.
Let's say we have $a$, $b$, and $c$ such that $a^b = c$.
To rearrange this equation for $a$, we can use a radical, making $a = \sqrt[b]{c}$.
But if we want to rearrange for $b$ instead, we have to use a logarithm. This equation looks like:
$$b = \log_a c$$
So essentially, if you have $\log_a c$, it's asking "What number do I have to raise $a$ to in order to make $c$?
ExampleIf $2^3 = 8$, then $\log_2 8 = 3$ If $10^x = 1000$, then $\log_{10} 1000 = x$ (which we know is 3)
Base 10 Logarithms
The base 10 logarithm, often written as $\log_{10}$ or simply $\log$, is the most commonly used logarithm in everyday applications. It represents the power to which 10 must be raised to obtain a given number.
Example$\log_{10} 100 = 2$ because $10^2 = 100$ $\log_{10} 1000 = 3$ because $10^3 = 1000$
Natural Logarithms (Base $e$)
The natural logarithm, denoted as $\ln$ or $\log_e$, uses the mathematical constant e (approximately 2.71828) as its base. Natural logarithms are particularly important in calculus and many scientific applications.
NoteIt's crucial to remember that $\log_e x = \ln x$. These notations are equivalent and both refer to the natural logarithm.
Numerical Evaluation of Logarithms
In practice, logarithms are often evaluated using technology such as scientific calculators or computer software. This means a GDC when you're taking your exams. This is because exact values of logarithms are often irrational numbers.
ExampleUsing a calculator: $\log_{10} 7 \approx 0.8451$ $\ln 7 \approx 1.9459$
TipWhen using a calculator, make sure you're using the correct logarithm function (log for base 10, ln for natural logarithm) as per the question requirements.
