How to Understand Logarithms
Know the difference between logarithmic and exponential equations., Know the parts of a logarithm., Know the difference between a common log and a natural log., Know and apply the properties of logarithms., Practice using the properties.
Step-by-Step Guide
-
Step 1: Know the difference between logarithmic and exponential equations.
This is a very simple first step.
If it contains a logarithm (for example: logax = y) it is logarithmic problem.
A logarithm is denoted by the letters "log".
If the equation contains an exponent (that is, a variable raised to a power) it is an exponential equation.
An exponent is a superscript number placed after a number.
Logarithmic: logax = y Exponential: ay = x -
Step 2: Know the parts of a logarithm.
The base is the subscript number found after the letters "log"--2 in this example.
The argument or number is the number following the subscript number--8 in this example.
Lastly, the answer is the number that the logarithmic expression is set equal to--3 in this equation., Common logs have a base of
10. (for example, log10x).
If a log is written without a base (as log x), then it is assumed to have a base of
10.
Natural logs:
These are logs with a base of e. e is a mathematical constant that is equal to the limit of (1 + 1/n)n as n approaches infinity, approximately
2.718281828. (It has many more digits than those written here.) logex is often written as ln x.
Other Logs:
Other logs have the base other than that of the common log and the E mathematical base constant.
Binary logs have a base of 2 (for the example, log2x).
Hexadecimal logs have the base of 16 (for the example log16x (or log#0fx in the notation of hexadecimal).
Logs that have the 64th base are indeed quite complex, and therefore are usually restricted to the Advanced Computer Geometry (ACG) domain. , The properties of logarithms allow you to solve logarithmic and exponential equations that would be otherwise impossible.
These only work if the base a and the argument are positive.
Also the base a cannot be 1 or
0.
The properties of logarithms are listed below with a separate example for each one with numbers instead of variables.
These properties are for use when solving equations. loga(xy) = logax + logay A log of two numbers, x and y, that are being multiplied by each other can be split into two separate logs: a log of each of the factors being added together. (This also works in reverse.) Example: log216 = log28*2 = log28 + log22 loga(x/y) = logax
- logay A log of a two numbers being divided by each other, x and y, can be split into two logs: the log of the dividend x minus the log of the divisor y.
Example: log2(5/3) = log25
- log23 loga(xr) = r*logax If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.
Example: log2(65) 5*log26 loga(1/x) =
-logax Think about the argument. (1/x) is equal to x-1.
Basically this is another version of the previous property.
Example: log2(1/3) =
-log23 logaa = 1 If the base a equals the argument a the answer is
1.
This is very easy to remember if one thinks about the logarithm in exponential form.
How many times should one multiply a by itself to get a? Once.
Example: log22 = 1 loga1 = 0 If the argument is one the answer is always zero.
This property holds true because any number with an exponent of zero is equal to one.
Example: log31 =0 (logbx/logba) = logax This is known as "Change of Base".One log divided by another, both with the same base b, is equal to a single log.
The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument.
This is easy to remember if you think about the base as the bottom of an object and the denominator as the bottom of a fraction.
Example: log25 = (log 5/log 2) , These properties are best memorized by repeated use when solving equations.
Here's an example of an equation that is best solved with one of the properties: 4x*log2 = log8 Divide both sides by log2. 4x = (log8/log2) Use Change of Base. 4x = log28 Compute the value of the log. 4x = 3 Divide both sides by
4. x = 3/4 Solved.
This is very helpful.
I now understand logs. -
Step 3: Know the difference between a common log and a natural log.
-
Step 4: Know and apply the properties of logarithms.
-
Step 5: Practice using the properties.
Detailed Guide
This is a very simple first step.
If it contains a logarithm (for example: logax = y) it is logarithmic problem.
A logarithm is denoted by the letters "log".
If the equation contains an exponent (that is, a variable raised to a power) it is an exponential equation.
An exponent is a superscript number placed after a number.
Logarithmic: logax = y Exponential: ay = x
The base is the subscript number found after the letters "log"--2 in this example.
The argument or number is the number following the subscript number--8 in this example.
Lastly, the answer is the number that the logarithmic expression is set equal to--3 in this equation., Common logs have a base of
10. (for example, log10x).
If a log is written without a base (as log x), then it is assumed to have a base of
10.
Natural logs:
These are logs with a base of e. e is a mathematical constant that is equal to the limit of (1 + 1/n)n as n approaches infinity, approximately
2.718281828. (It has many more digits than those written here.) logex is often written as ln x.
Other Logs:
Other logs have the base other than that of the common log and the E mathematical base constant.
Binary logs have a base of 2 (for the example, log2x).
Hexadecimal logs have the base of 16 (for the example log16x (or log#0fx in the notation of hexadecimal).
Logs that have the 64th base are indeed quite complex, and therefore are usually restricted to the Advanced Computer Geometry (ACG) domain. , The properties of logarithms allow you to solve logarithmic and exponential equations that would be otherwise impossible.
These only work if the base a and the argument are positive.
Also the base a cannot be 1 or
0.
The properties of logarithms are listed below with a separate example for each one with numbers instead of variables.
These properties are for use when solving equations. loga(xy) = logax + logay A log of two numbers, x and y, that are being multiplied by each other can be split into two separate logs: a log of each of the factors being added together. (This also works in reverse.) Example: log216 = log28*2 = log28 + log22 loga(x/y) = logax
- logay A log of a two numbers being divided by each other, x and y, can be split into two logs: the log of the dividend x minus the log of the divisor y.
Example: log2(5/3) = log25
- log23 loga(xr) = r*logax If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.
Example: log2(65) 5*log26 loga(1/x) =
-logax Think about the argument. (1/x) is equal to x-1.
Basically this is another version of the previous property.
Example: log2(1/3) =
-log23 logaa = 1 If the base a equals the argument a the answer is
1.
This is very easy to remember if one thinks about the logarithm in exponential form.
How many times should one multiply a by itself to get a? Once.
Example: log22 = 1 loga1 = 0 If the argument is one the answer is always zero.
This property holds true because any number with an exponent of zero is equal to one.
Example: log31 =0 (logbx/logba) = logax This is known as "Change of Base".One log divided by another, both with the same base b, is equal to a single log.
The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument.
This is easy to remember if you think about the base as the bottom of an object and the denominator as the bottom of a fraction.
Example: log25 = (log 5/log 2) , These properties are best memorized by repeated use when solving equations.
Here's an example of an equation that is best solved with one of the properties: 4x*log2 = log8 Divide both sides by log2. 4x = (log8/log2) Use Change of Base. 4x = log28 Compute the value of the log. 4x = 3 Divide both sides by
4. x = 3/4 Solved.
This is very helpful.
I now understand logs.
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Charlotte Ward
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