Logarithm

Welcome to Logarithm.app. On this site you can find everything about logarithms.

Every inverse function f−1(x) is a function that undoes another function f(x).

It essentially means that if an input x of the function f(x) produces y as output, then the input y of the inverse function f−1(x) produces the output x. And vice versa!

Logarithm are the inverse functions to exponentiations.

log

Result:

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Logarithm Definition

by = x ⇔ y = logb(x)

b, x, y ∈ R+, b ≠ 1

R+ denotes the set of positive real numbers.

Logarithm Parts

b is the base
x is the argument
y is the exponent

Logarithm Examples

Logarithmic FormExponential form
log2(8) = 323 = 8
log10(1000) = 3103 = 1000
ln(x) = 1e0 = 1
log3(9) = 233 = 9
log(100) = 2102 = 100

ln(x) is the natural logarithm which uses the number e as base. You may think of it as loge(x).

The logarithm with base 2 is called binary logarithm, and the logarithm with base 10 is usually called decimal logarithm.

The decimal logarithm, which is also known as common logarithm and decadic logarithm, often has it’s base omitted:

log10(x) = log(x)

The graph below depicts ln(x), log2(x) as well as log(x) for small values of x.

Evaluating Logarithms

1. Assumed you want to solve this equation:

log10(1000000) = y.

Write it in the equivalent form:

10y = 1000000

Take the log with base 10 from both sides:

log1010y = log10(1000000)

y = 6

2. Supposed you want to solve:

log2(32) = y.

Write it in the equivalent form:

2y = 32

Take the log with base 2 from both sides:

log22y = log2(32)

y = 5

Practice makes experts, so you may soon find yourself skipping the procedure and evaluating a log just by asking, “b to what power is y”?

3. Try solving log4(64) by asking yourself “4 to what power is 64?”