The exact value of log 100 base 2 is 2 / log₁₀(2). This can be approximated as 6.6438.
A logarithm answers the question: "To what power must a base be raised to produce a given number?" In mathematical terms, if we have log_b(a) = x, it means that b raised to the power of x equals a (i.e., b^x = a).
The common logarithm, sometimes referred to as the decimal logarithm, uses 10 as its base. It is frequently denoted as log x without an explicit base. For instance, the common logarithm of 100 (log 100) equals 2, because 10 raised to the power of 2 gives 100 (10² = 100). This property is crucial for calculating log 100 base 2.
Calculating the Exact Value of log₂100
To find the value of a logarithm with a less common base, such as base 2 for the number 100, we typically use the change of base formula. This formula allows us to convert a logarithm from one base to another, usually to base 10 (common logarithm) or base e (natural logarithm), which are readily available on calculators.
The change of base formula is:
log_b(a) = log_c(a) / log_c(b)
Where:
bis the original base (in our case, 2)ais the number (in our case, 100)cis the new base (we'll use 10)
Applying the formula to log₂100:
- Set
a = 100,b = 2, andc = 10. log₂(100) = log₁₀(100) / log₁₀(2)- As established,
log₁₀(100)(or simplylog 100) is equal to 2. - Therefore,
log₂(100) = 2 / log₁₀(2)
This fractional expression, 2 / log₁₀(2), represents the exact answer. If you need a numerical approximation, log₁₀(2) is approximately 0.30103.
So, log₂(100) ≈ 2 / 0.30103 ≈ 6.643856.
Understanding Different Logarithm Bases
Logarithms are used across various fields, and their base is often chosen based on the context:
- Common Logarithm (Base 10): Denoted as
log xorlog₁₀x. Widely used in chemistry (pH scale), physics (decibels), and engineering. - Natural Logarithm (Base e): Denoted as
ln xorlog_ex. Fundamental in calculus, financial mathematics, and exponential growth/decay models. The constant e is an irrational number approximately equal to 2.71828. - Binary Logarithm (Base 2): Denoted as
log₂x. Predominantly used in computer science, information theory, and digital signal processing, where values are often represented in binary (0s and 1s).
Key Properties of Logarithms
Understanding logarithm properties is essential for solving more complex logarithmic equations and expressions.
| Property | Rule | Example |
|---|---|---|
| Product Rule | log_b(xy) = log_b(x) + log_b(y) |
log₂(8) = log₂(2 * 4) = log₂(2) + log₂(4) = 1 + 2 = 3 |
| Quotient Rule | log_b(x/y) = log_b(x) - log_b(y) |
log₂(2) = log₂(4/2) = log₂(4) - log₂(2) = 2 - 1 = 1 |
| Power Rule | log_b(x^n) = n * log_b(x) |
log₂(64) = log₂(2⁶) = 6 * log₂(2) = 6 * 1 = 6 |
| Change of Base | log_b(a) = log_c(a) / log_c(b) |
log₂(100) = log₁₀(100) / log₁₀(2) |
For further exploration of logarithm concepts and properties, you can refer to resources like Khan Academy on Logarithms or Wikipedia's page on Logarithms.
Practical Applications of Logarithms
Logarithms simplify complex calculations and describe phenomena that span wide ranges of values. Some practical applications include:
- Sound Intensity: Measured in decibels (dB), which use a logarithmic scale to represent the vast range of sound pressures audible to the human ear.
- Earthquake Magnitude: The Richter scale, used to measure earthquake intensity, is a base-10 logarithmic scale.
- pH Scale: In chemistry, the pH scale, which measures the acidity or alkalinity of a solution, is logarithmic, based on the concentration of hydrogen ions.
- Computer Science: Logarithms are fundamental in analyzing the efficiency of algorithms, especially in sorting and searching, and in information theory (e.g., bits are log base 2).
- Financial Growth: Used in calculating compound interest and modeling financial growth over time.
