How to Put Log Base 4 with A Scientific Calculator

Calculating logarithms with base 4 is a common mathematical operation that appears in various fields including computer science, engineering, and finance. This guide explains how to perform log base 4 calculations using a scientific calculator and provides a step-by-step method.

How to Calculate Log Base 4

The logarithm base 4 of a number x, written as log₄x, is the exponent to which the base 4 must be raised to obtain x. Mathematically, this is expressed as:

log₄x = y if and only if 4ʸ = x

To calculate log₄x, you can use the change of base formula which allows you to use a calculator that only has log₁₀ (common logarithm) and ln (natural logarithm) functions.

Using a Scientific Calculator

Most scientific calculators have a log function that calculates the common logarithm (base 10). To calculate log base 4, you can use the change of base formula:

log₄x = log₁₀x / log₁₀4

Here's how to perform the calculation:

Enter the number x you want to calculate the logarithm of.
Press the log button to calculate log₁₀x.
Enter the number 4.
Press the log button again to calculate log₁₀4.
Divide the first result by the second result to get log₄x.

Note: Some calculators may have a "log" button that calculates natural logarithm (ln). If your calculator has this, you'll need to use the change of base formula with natural logarithms: log₄x = lnx / ln4.

The Formula

The change of base formula for logarithms is a fundamental tool in mathematics. It allows you to calculate logarithms with any base using a calculator that only has base 10 and natural logarithm functions. The formula is:

logₐb = logₖb / logₖa

Where:

a is the desired base (4 in our case)
b is the number you want to calculate the logarithm of
k is the base of the calculator's logarithm function (10 or e)

This formula works because logarithms with different bases are proportional to each other. The change of base formula is valid for any positive real numbers a, b, and k where a ≠ 1 and k ≠ 1.

Worked Example

Let's calculate log₄16 using a scientific calculator that has base 10 logarithm function.

Enter 16 and press the log button: log₁₀16 ≈ 1.2041
Enter 4 and press the log button: log₁₀4 ≈ 0.6021
Divide the first result by the second: 1.2041 / 0.6021 ≈ 2

The result is 2, which means log₄16 = 2 because 4² = 16.

Verification: 4² = 4 × 4 = 16, which confirms our calculation is correct.

FAQ

Can I calculate log base 4 without a calculator?

Yes, you can use the change of base formula with logarithms you know or can calculate manually. However, this method is more complex and typically requires a calculator for intermediate steps.

What if my calculator doesn't have a log button?

If your calculator only has natural logarithm (ln) function, you can use the change of base formula with natural logarithms: log₄x = lnx / ln4. If your calculator has neither, you'll need to use the exponential function and trial and error.

Is log base 4 the same as log base 2?

No, log base 4 and log base 2 are different functions. For example, log₄16 = 2 while log₂16 = 4. The relationship between them can be expressed using the change of base formula: log₄x = log₂x / log₂4 = log₂x / 2.