Desmos Calculator How to Put Base in Log

Changing the base of a logarithm in Desmos calculator can be tricky if you're not familiar with the change of base formula. This guide will walk you through the process step-by-step, including how to implement it in Desmos and why you might need to do this in the first place.

How to Change the Base of a Logarithm in Desmos

Desmos is a powerful graphing calculator that supports logarithmic functions, but it doesn't have a built-in function to change the base of a logarithm directly. Instead, you'll need to use the change of base formula to convert between different logarithmic bases.

Remember that Desmos uses natural logarithms (ln) and common logarithms (log) by default. The change of base formula allows you to convert between any bases.

To change the base of a logarithm in Desmos, you'll need to:

Identify the original logarithm and its base
Determine the desired target base
Apply the change of base formula
Implement the formula in Desmos

Why You Might Need to Change the Logarithm Base

There are several reasons why you might need to change the base of a logarithm:

Different scientific fields use different logarithmic bases (common log base 10, natural log base e)
You need to compare logarithms with different bases in the same graph
You're working with logarithmic scales that require a specific base
You're converting between different logarithmic representations

Understanding how to change the base of a logarithm is essential for working with logarithmic functions in Desmos and other graphing calculators.

The Change of Base Formula

The change of base formula allows you to convert a logarithm from one base to another. The formula is:

log_b(x) = log_k(x) / log_k(b)

Where:

log_b(x) is the logarithm of x with base b (the target base)
log_k(x) is the logarithm of x with base k (the original base)
log_k(b) is the logarithm of b with base k

In practice, you can use natural logarithms (ln) or common logarithms (log) as the base k, as they are available in most calculators including Desmos.

Step-by-Step Guide to Changing Logarithm Base in Desmos

Step 1: Identify the Original Logarithm

First, identify the logarithm you want to convert. For example, if you have log₂(x), this is your original logarithm with base 2.

Step 2: Choose a Target Base

Decide what base you want to convert to. Common choices are base 10 (common logarithm) or base e (natural logarithm).

Step 3: Apply the Change of Base Formula

Using the change of base formula, express your original logarithm in terms of the target base. For example, to convert log₂(x) to base 10:

log₁₀(x) = log₂(x) / log₂(10)

Step 4: Implement in Desmos

In Desmos, you can implement this conversion by:

Creating a slider for x
Defining the original logarithm: y = log₂(x)
Defining the converted logarithm using the change of base formula
Plotting both functions on the same graph

Here's an example of how you might set this up in Desmos:

1. Create a slider: x = 1..100

2. Define original log: y1 = log(x)/log(2)

3. Define converted log: y2 = log(x)/log(10)

4. Plot y1 and y2 on the same graph

Common Mistakes When Changing Logarithm Bases

When changing logarithm bases, it's easy to make a few common mistakes:

Forgetting to divide by log_k(b) in the change of base formula
Mixing up the numerator and denominator in the formula
Using the wrong base k (natural log vs. common log)
Not simplifying the expression properly

Double-check your calculations and verify your results with a calculator to avoid these errors.

Practical Examples of Logarithm Base Conversion

Example 1: Converting from Base 2 to Base 10

Convert log₂(8) to base 10:

log₁₀(8) = log₂(8) / log₂(10) = 3 / log₂(10)

≈ 3 / 3.3219 ≈ 0.9031

Example 2: Converting from Base e to Base 10

Convert ln(10) to base 10:

log₁₀(10) = ln(10) / ln(10) = 1

Example 3: Converting from Base 10 to Base 2

Convert log₁₀(100) to base 2:

log₂(100) = log₁₀(100) / log₁₀(2) = 2 / 0.3010 ≈ 6.6439

Frequently Asked Questions

Can I change the base of a logarithm directly in Desmos?

No, Desmos doesn't have a built-in function to change the base of a logarithm directly. You need to use the change of base formula.

What's the difference between natural log and common log?

Natural log (ln) uses base e (approximately 2.71828), while common log (log) uses base 10. They're both useful in different contexts.

How do I know which base to use for my calculations?

The choice of base depends on your specific application. Common log (base 10) is often used in engineering and science, while natural log (base e) is common in calculus and statistics.

Is there a simpler way to change logarithm bases in Desmos?

The change of base formula is the standard method, but you can create custom functions in Desmos to simplify the process for repeated use.