The basic rules of exponents state that some number ( a ) raised to the ( x ) power is equal to ( y ). For example, ( 2 )( 2 ) = 4. This can also be expressed as ” two squared ” = 2^2. Likewise, ( 2 )( 2 )( 2 ) = 2^3 = 8. Therefore, a^x = y, where ( x ) is some value that is chosen, and ( y ) is the consequence of the value of ( x ). If 2^x = y, and we choose x = 4, the statement that a^x = y becomes 2^4 = 16, where a = 2, x = 4, and the consequence of those choices is y = 16.
The logarithmic function ( log x = y ) is the mathematically opposite process of ( a^x = y ). The goal is to find out what exponent would be needed to convert some base number into another value. For example, if 10^2 = ( 10 )( 10 ) = 100, the value of 2 was the exponent needed to raise ( or convert ) the base value of 10 to equal 100. The logarithmic ( log10 ) function is written as log x = y. Here, 10 is omitted for convenience. If x = 100 in log x = y, then log 100 = 2, because it would take an exponent of 2 to convert the value of 10 into 100. If 10^2 = 100, then log 100 = 2. log 1000 = 3, because ( 10 )( 10 )( 10 ) = ( 10^3 ) = 1,000.
The rules used to multiply and divide exponential numbers can be used to further expand the rules of logarithms. In general, ( a^2 )( a^3 ) = ( a )( a )( a )( a )( a ) = a^5. If a = 2, then ( 2^2 )( 2^3 ) = ( 2 )( 2 )( 2 )( 2 )( 2 ) = 2^5 = 32. Therefore, ( 2^x )( 2^y ) = 2^( x + y ). When one exponential number is divided by another, it can be proven that ( a^x / a^y ) = a^( x – y ).
The log ( a^x / a^y ) is a way of stating that the process of dividing ( a^x / a^y ) is achieved by subtracting the exponents ( x ) and ( y ). For example, if we divide ( 2^2 ) by ( 2^3 ), we can analytically see that ( 4/8 ) = ( 1/2 ). Similarly, ( 2^( 2 – 3 ) ) = ( 2^-1 ) = ( 1/2 ). Therefore, if ( a^x / a^y ) = a^( x – y ), then log ( a^x / a^y ) = log ( a^x ) – log ( a^y ). Likewise, if ( a^x )( a^y ) = a^( x + y ), the statement that the log ( ( a^x)( a^y ) ) = log ( a^x ) + log ( a^y ) confirms that the process of multiplying ( a^x ) with ( a^y ) is achieved by adding the exponents together. If ( 2^3 ) is multiplied by ( 2^2 ), we have ( 2 )( 2 )( 2 )( 2 )( 2 ) = ( 2^5 ) = 32. Likewise, the log ( ( 10^2 )( 10^3 ) ) = log ( 10^2 ) + log ( 10^3 ) = 2 + 3 = 5. Therefore, a base of 10 would have to be raised by a factor of 5 to achieve the result of multiplying ( 10^2 )( 10^3 ). Log ( ( 10^2 )( 10^3 ) ) = 5.
A thorough understanding of the rules of logs is crucial in the fields of mathematics, chemistry, physics, and biology. Logarithms are oftentimes used to describe mathematical processes that are broad in scope and would therefore be cumbersome to graph or interpret in a meaningful way. For example, the pH of a solution is interpreted as being the – log ( + H ), where ( + H ) is a concentration of hydrogen ions. The concentration of hydrogen ions ( + H ) in a pure sample of water is 10^-7. Therefore, a base number of 10 would need to be decreased by a factor of -7 to describe the concentration of ( + H ) in purified water. Therefore, log ( + H ) of a relatively ” neutral ” solution of water = -7. Our preference here is to work with positive numbers, so – log ( + H ) = 7 for a pure sample of water.
Mathematically, a pH value of 6 means that a hydrogen ion concentration is 10 times greater than ” neutral ” water ( 10^-6 vs 10^-7 ). A solution with a pH value that is lower than 7 is referred to as being ” acidic “. A solution that has a pH value that is greater than 7 is referred to as being ” basic “. Therefore, if a solution has a pH value = 8, the hydrogen ion concentration is 10^-8. This value is one-tenth that of 10^-7.
Similar mathematical operations are extensively used to examine the phenomena of sound. The decibel ( B ) is the logarithmic function used to quantify how sound is perceived by the human ear.
RENAISSANCE PHYSICS 