How Do You Find the Value of a Log?

Finding the value of a logarithm depends on whether you're working with common logarithms (base 10), natural logarithms (base e), or logarithms with a different base. Here's a breakdown of the methods:

1. Common Logarithms (Base 10)

These are the most frequently encountered logs and are often written as "log(x)" with no base explicitly stated (implying base 10).
Using a Calculator: Most scientific calculators have a "log" button. To find log(x), simply enter x and press the "log" button. For example, to find log(100), enter 100 and press "log". The calculator will display 2, because 10² = 100.

2. Natural Logarithms (Base e)

Natural logarithms are written as "ln(x)" and have a base of e (Euler's number, approximately 2.71828).
Using a Calculator: Scientific calculators also have an "ln" button. To find ln(x), enter x and press the "ln" button. For example, to find ln(20), enter 20 and press "ln". The calculator will display approximately 2.9957. This means e^2.9957 ≈ 20.

3. Logarithms with a Different Base

When the base is not 10 or e, you'll need to use the change of base formula. This formula allows you to convert a logarithm of any base to a logarithm of base 10 or base e, which your calculator can then evaluate.

The Change of Base Formula:

log_b(a) = log_c(a) / log_c(b)

Where:

log_b(a) is the logarithm you want to find.
b is the base of the logarithm you want to find.
a is the argument of the logarithm you want to find.
c is the new base (usually 10 or e).

Example:

Let's say you want to find log₂(16) (log base 2 of 16).

Apply the change of base formula: log₂(16) = log₁₀(16) / log₁₀(2)
Use a calculator to find the common logarithms: log₁₀(16) ≈ 1.2041 and log₁₀(2) ≈ 0.3010
Divide: 1.2041 / 0.3010 ≈ 4

Therefore, log₂(16) = 4, because 2⁴ = 16.

Alternative Change of Base Formula using Natural Logarithms:

You can also use natural logarithms:

log_b(a) = ln(a) / ln(b)

Using the same example, log₂(16):

log₂(16) = ln(16) / ln(2)
ln(16) ≈ 2.7726 and ln(2) ≈ 0.6931
2.7726 / 0.6931 ≈ 4

4. Understanding the Logarithmic Equation

The fundamental relationship between logarithms and exponents is:

log_b(x) = y is equivalent to b^y = x
This understanding helps you to conceptualize what a logarithm represents: "To what power must I raise the base b to get x?" The answer is y.

Summary:

To find the value of a logarithm, use a calculator directly for common (base 10) and natural (base e) logs. For other bases, apply the change of base formula to convert to base 10 or base e before using your calculator. Remember the relationship between logarithms and exponents to better understand the results.

askvity