How to Understand Logarithms: 5 Steps (with Pictures)

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

Confused by logarithms? Do not worry! A logarithm (abbreviated log) is nothing more than an exponent in a different form.

log_tox = y is the same as a^y = x.

Steps

Step 1. Know the difference between logarithmic and exponential equations

It is a very simple step. If it contains a logarithm (for example: log_tox = y) is a logarithmic problem. A logarithm is represented by letters "log"If the equation contains an exponent (which is a variable raised to a power), then it is an exponential equation. An exponent is a superscript number after another number.

• Logarithmic: log_tox = y
• Exponential: a^y = x

Step 2. Learn the parts of a logarithm

The base is the number subscribed after the letters "log" - 2 in this example. The argument or number is the number following the subscribed number - 8 in this example. The result is the number that the logarithmic expression puts equal to - 3 in this equation.

Step 3. Know the difference between a common logarithm and a natural logarithm

• common log: are base 10 (for example, log₁₀x). If a logarithm is written without the base (such as log x), then the base is assumed to be 10.
• natural log: are logarithms to the base e. e is a mathematical constant which is equal to the limit of (1 + 1 / n) with n tending towards infinity, approximately 2, 718281828. (has many more digits than given here) log_Andx is often written as ln x.
• Other logarithms: other logarithms have a base other than 10 and e. Binary logarithms are base 2 (for example, log₂x). Hexadecimal logarithms are base 16 (e.g. log₁₆x or log_{# 0f}x in hexadecimal notation). Logarithms to base 64^th they are very complex, and usually restricted to very advanced geometry calculations.

Step 4. Know and apply the properties of logarithms

The properties of logarithms allow you to solve logarithmic and exponential equations otherwise impossible to solve. They only work if the base a and the argument are positive. Also the base a cannot be 1 or 0. The properties of the logarithms are listed below with an example for each of them, with numbers instead of variables. These properties are useful for solving equations.

• log_to(xy) = log_tox + log_toy

  A logarithm of two numbers, x and y, which are multiplied by each other, can be divided into two separate logs: a log of each of the factors added together (it also works in reverse).

  Example:

  log₂16 =

  log₂8*2 =

  log₂8 + log₂2

• log_to(x / y) = log_tox - log_toy

  A log of two numbers divided by each of them, x and y, can be divided into two logarithms: the log of the dividend x minus the log of the divisor y.

  example:

  log₂(5/3) =

  log₂5 - log₂3

• log_to(x^r) = r * log_tox

  If the log argument x has an exponent r, the exponent can be shifted in front of the logarithm.

  Example:

  log₂(6⁵)

  5 * log₂6

• log_to(1 / x) = -log_tox

  Look at the topic. (1 / x) equals x^-1. This is another version of the previous property.

  Example:

  log₂(1/3) = -log₂3

• log_toa = 1

  If the base a is equal to the argument a, the result is 1. This is very easy to remember if you think of the logarithm in exponential form. How many times would you have to multiply a by itself to get a? Once.

  Example:

  log₂2 = 1

• log_to1 = 0

  If the argument is 1, the result is always 0. This property is true because any number with an exponent of 0 equals 1.

  Example:

  log₃1 =0

• (log_bx / log_ba) = log_tox

  This is known as "base change". One logarithm divided by another, both with the same base b, equals the single logarithm. The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument. It's easy to remember if you think of the base as the base of an object and the denominator as the base of a fraction.

  Example:

  log₂5 = (log 5 / log 2)

Step 5. Practice with the properties

Properties are stored by practicing solving equations. Here is an example of an equation that can be solved with one of the properties:

4x * log2 = log8 divide both by log2.

4x = (log8 / log2) Use base change.

4x = log₂8 Compute the value of log.4x = 3 Divide both by 4. x = 3/4 End.