How to Put Log Base in Desmos Calculator

Desmos is a powerful graphing calculator that can handle logarithms with custom bases. This guide shows you how to input and work with logarithmic functions in Desmos, including how to change the base of the logarithm.

Introduction

Logarithms are mathematical functions that solve for the exponent in an exponential equation. The basic logarithm function is written as log_b(x), where b is the base and x is the argument. In Desmos, you can work with logarithms using the built-in log function, which defaults to base 10.

This guide will show you how to:

Input basic logarithms in Desmos
Change the base of the logarithm
Understand the relationship between different logarithmic bases
Create graphs of logarithmic functions

Basic Logarithm in Desmos

To input a basic logarithm in Desmos:

Open the Desmos graphing calculator at https://www.desmos.com/calculator
In the expression list on the left side, type y = log(x)
Press Enter

This will plot the base-10 logarithm function. The graph will show the curve of log₁₀(x).

The log function in Desmos defaults to base 10. If you want to use natural logarithms (base e), you can use ln(x).

Setting a Custom Base

Desmos doesn't have a built-in function for logarithms with arbitrary bases, but you can create them using the change of base formula:

log_b(x) = ln(x)/ln(b)

To create a logarithm with a custom base in Desmos:

In the expression list, type y = ln(x)/ln(2) for a base-2 logarithm
Press Enter
This will plot log₂(x)

You can change the base by modifying the second ln function. For example:

y = ln(x)/ln(3) for base-3 logarithm
y = ln(x)/ln(5) for base-5 logarithm
y = ln(x)/ln(10) for base-10 logarithm (equivalent to the built-in log function)

Remember that the base must be a positive real number greater than 0 and not equal to 1. The argument x must also be positive.

Worked Example

Let's create a base-5 logarithm in Desmos and evaluate it at x = 625.

In the expression list, type y = ln(x)/ln(5)
Press Enter
In the expression list, type (625, ln(625)/ln(5)) to create a point at x=625
Press Enter

The point will appear at (625, 4) because 5⁴ = 625. This confirms that log₅(625) = 4.

You can also create a slider to dynamically change the base. Type b = 2 and then y = ln(x)/ln(b). The graph will update as you adjust the slider.

FAQ

Can I use any base with the logarithm function in Desmos?: Yes, you can use any positive real number as the base, except for 1. The change of base formula allows you to create logarithms with any valid base.
How do I graph multiple logarithmic functions with different bases?: You can add multiple expressions to the expression list, each with a different base. For example, y = ln(x)/ln(2), y = ln(x)/ln(3), and y = ln(x)/ln(5).
Can I use logarithms with complex numbers in Desmos?: Desmos does not support complex numbers in its basic logarithm function. The logarithm function in Desmos is defined only for positive real numbers.
How do I evaluate a logarithm at a specific point?: You can create a point by typing (x, ln(x)/ln(b)) where x is the point you want to evaluate and b is the base. The y-coordinate will be the value of the logarithm.
Is there a way to simplify logarithmic expressions in Desmos?: Desmos does not have built-in tools for simplifying logarithmic expressions, but you can manually simplify expressions using the logarithm properties and the change of base formula.